Did the Pilgrims build log cabins? Nope. "Neither then nor later did the pilgrims build log cabins for the good reason that they did not know how."
(Believe it or not, Scandinavian immigrants arriving about 1640 to settlements along the Delaware, brought the know-how to erect log cabins.)
Most spent that winter of 1620-1621 still onboard the Mayflower (at least it was shelter). As soon as spring finally came, they began building a few small cottages of wattle and daub construction with steep thatched roofs, typically English in design. Each family was to build its own house and each family to take in one or more of the single persons. But it was not until March 21st that the last of the Pilgrims finally left the Mayflower and "came ashore with much adoe to live henceforth on ye firme and stable earth, their proper elemente." It had been eight long wintery months since the ship first sailed into Provincetown harbor.
I've been blessed to visit Plimouth Plantation. These replica "cottages" are not someplace where you or I would want to live. They were dirt floored, tiny in area, windowless, crowded with living and with people. Yet they did survive; I am a descendant of nine passengers who were on the Mayflower, both Saints and Strangers.
(Quoting from Saints and Strangers, by George F. Willison, 1945.)