We sailed into Iceland three times and never had any kind of immigration check. I've asked about it in Seydisfjordur and Iceland and received odd looks from the visitor info desk operators who seem shocked that I would even ask the question (I'm a keen collector of passport stamps).
So no, it is extremely unlikely that you will have to go through immigration unless you are travelling after the roll-out of ETIAS throughout the EU. When that happens, all bets are off.
https://www.journeyintodarkness.co.uk/category/iceland