Here is the item again:
Searching travel websites
(A) to compare room
and flight rates,
(B) Shaquita
fantasized about a spring break vacation to
Hawaii. (C)
As her unfinished algebra homework lay ignored on her desk.
You correctly realized that the period after Hawaii created a fragment. As her unfinished algebra homework lay ignored ... is a subordinate clause, which cannot stand alone as a complete sentence.
To fix this fragment, you chose to do this:
Searching travel websites to compare room and flight rates,
Shaquita fantasized about a spring break vacation to
Hawaii as her unfinished algebra homework lay
ignored on her desk.
You have correctly employed Punctuation Rule 1:
Main Clause + Ø + Subordinate Clause.
Rock on!