They spent nine months on the island, during which time they constructed two new ships out of Bermuda timber and pieces taken from the Sea Venture. Some settlers died while on the island, and two men were intentionally left on the island for unspecified crimes.
In 1610, survivor Sylvester Jordain published a book entitled A discovery of the Bermudas, otherwise called the Isle of Devils, which apparently included an account of the storm and the wreck. That same year, William Strachey wrote a letter detailing the wreck and what followed (including his time in Jamestown). The letter was published in 1625 as A true reportory of the wracke, and redemption of Sir THOMAS GATES Knight, but the Jordain book would have provided some of the same information as Strachey's account of the storm and the wreck - to say nothing of everyone surviving it, at least in the first instance, and then living to not only tell the tale, but also to sail off to their original destination (where they found nothing but death and the dying at Jamestown, but that's a different story) - definitely match up with some of The Tempest's plot points. I do not believe it likely that Shakespeare saw Strachey's letter, but the information in its contents tells some of the story of what transpired aboard the Sea Venture.
Here's a (relatively) short excerpt from Strachey's letter to the "Excellent Lady". A longer portion of the letter is available on the Folger Shakespeare Library website.
We had followed this course so long, as now we were within seven or eight days at the most, by Captain Newport’s reckoning, of making Cape Henry upon the coast of Virginia: When on St. James his day, July 24, being Monday (preparing for no less all the black night before) the clouds gathering thick upon us, and the winds singing, and whistling most unusually, which made us to cast off our Pinnace, towing the same until then astern, a dreadful storm and hideous began to blow from out the Northeast, which swelling, and roaring as it were by fits, some hours with more violence then others, at length did beat all light from heaven; which like an hell of darkness turned black upon us, so much the more fuller of horror, as in such cases horror and fear use to overrun the troubled, and overmastered senses of all, which (taken up with amazement) the ears lay so sensible to the terrible cries, and murmurs of the winds, and distraction of our Company, as who was most armed, and best prepared, was not a little shaken. For surely (Noble Lady) as death comes not so sudden nor apparent, so he comes not so elvish and painful (to men especially even then in health and perfect habitudes of body) as at Sea; who comes at no time so welcome, but our frailty (so weak is the hold of hope in miserable demonstrations of danger) it makes guilty of many contrary changes, and conflicts: For indeed death is accompanied at no time, nor place with circumstances every way so uncapable of particularities of goodness and inward comforts, as at Sea. For it is most true, there ariseth commonly no such unmerciful tempest, compound of so many contrary and diverse Nations, but that it worketh upon the whole frame of the body , and most loathsomely affecteth all the powers thereof: and the manner of the sickness it lays upon the body, being so unsufferable, give not the mind any free and quiet time, to use her judgment and Empire...
For four and twenty hours the storm in a restless tumult, had blown so exceedingly, as we could not apprehend in our imaginations any possibility of greater violence, yet did we still find it, not only more terrible, but more constant, fury added to fury, and one storm urging a second more outrageous then the former; whether it so wrought upon our fears, or indeed met with new forces: Sometimes strikes in our Ship amongst women, and passengers, not used to such hurly and discomforts, made us look one upon the other with troubled hearts, and panting bosoms: our clamors drowned in the winds, and the winds in thunder. Prayers might well be in the heart and lips, but drowned in the outcries of the Officers: nothing heard that could give comfort, nothing seen that might encourage hope. It is impossible for me, had I the voice of Stentor, and expression of as many tongues, as his throat of voices, to express the outcries and miseries, not languishing, but wasting his spirits, and art constant to his own principles, but not prevailing. Our sails wound up lay without their use, and if at any time we bore but a Hollock, or half forecourse, to guide her before the Sea, six and sometimes eight men were not enough to hold the whipstaff in the steerage, and the tiller below in the Gunner room, by which may be imagined the strength of the storm: In which, the Sea swelled above the Clouds, and gave battle unto Heaven. It could not be said to rain, the waters like whole Rivers did flood in the air. And this I did still observe, that whereas upon the Land, when a storm hath powered itself forth once in drifts of rain, the wind as beaten down, and vanquished therewith, not long after endureth: here the glut of water (as if throttling the wind ere while) was no sooner a little emptied and qualified, but instantly the winds (as having gotten their mouths now free, and at liberty) spoke more loud, and grew more tumultuous, and malignant: What shall I say? Winds and Seas were as mad, as fury and rage could make them . . . there was not a moment in which the sudden splitting, or instant over-setting of the Ship was not expected.
. . .
Once so huge a Sea broke upon the poop and quarter, upon us, as it covered our Ship from stern to stem, like a garment or a vast cloud. It filled her brim full for awhile within, from the hatches up the sparre deck. . . . During all this time, the heavens looked so black upon us, that it was not possible the elevation of the Pole might be observed: nor a Star by night, not Sunbeam by day was to be seen. Only upon the Thursday night, Sir George Summers being upon the watch, had an apparition of a little round light, like a faint Star, trembling, and streaming along with a sparkling blaze, half the height upon the Main Mast, and shooting sometimes from Shroud to Shroud, attempting to settle as it were upon any of the four Shrouds; and for three or four hours together, or rather more, half the night it kept with us, running sometimes along the Mainyard to the very end, and then returning. At which, Sir George Summers called divers about him, and showed them the same, who observed it with much wonder, and carefulness: but upon a sudden, towards the morning watch, they lost the sight of it, and knew not what way it made. The superstitious Seamen make many constructions of this Sea-fire, which nevertheless is usual in storms: The same (it may be) which the Grecians were wont in the Mediterranean to call Castor and Pollux, of which, if one only appeared without the other, they took it for an evil sign of great tempest. The Italians, and such, who lie open to the Adriatic and Tyrrene Sea, call it (a sacred Body) Corpo sancto: the Spaniards call it Saint Elmo, and have an authentic and miraculous Legend for it.
. . . East and by South we steered away as much as we could to bear upright, which was no small carefulness nor pain to do, albeit we much unrigged our Ship, threw overboard much luggage, many a Trunk and Chest (in which I suffered no mean loss) and staved many a Butt of Beer, Hogsheads of Oil, Cider, Wine, and Vinegar, and heaved away all our Ordnance on the Starboard side, and had now purposed to have cut down the Main Mast, the more to lighten her, for we were much spent, and our men so weary, as their strengths together failed them, with their hearts, having travailed now from Tuesday till Friday morning, day and night, without either sleep or food.
The account of the severity and nature of the storm, and of St. Elmo's fire - which comports with Ariel's narration of how he appeared here and there upon the ship as flickers of light in Act I, scene 2 - certainly seem to have influenced Shakespeare, to say nothing of the "larger plot" wherein the ship is separated from the fleet, and nobody comes to any harm despite a storm so severe that even the seasoned sailors were panicked, only to have the survivors sail off again to rejoin the original fleet (although in the real world example, that was not exactly a happy ending).
There are some scholars who believe that Shakespeare's The Tempest drew its storm accounts from general shipwreck books and travel literature of the time, but I think it far more likely that the current event set him off, even if it's possible that he looked at Orlando Furioso by Ariosto (from 1516) and/or Naufragium (The Shipwreck) by Erasmus (from 1523), translated into English in 1606.