The Spanish plegar, meaning “to fold” comes from the Latin root plicare, meaning the same.
From plicare, we also get the English applicant. The connection makes sense if we think about both words in the sense of “attach”: when you apply, you want to attach yourself to an organization; and think of fold in the same metaphorical sense, “to bring into the fold.”
We can see the mapping clearly in the p‑l-g of plegar and the p‑l-c of applicant. The ‑c- was lost when it was shortened to just apply over time.
From the same root we also get the English ply, as in plywood — but that is a lot less common!