If I forgot to write a pruzbul am I permitted to collect my loan if the borrower is willing to pay me back?
There is a minority opinion that the laws of Shmittah nowadays are not absolutely binding but are rather an obligatory custom. See also Terumas Hadeshen (Responsum 304) as to whether the laws of Shemittas Kesafim apply nowadays outside Israel. If one forgot to arrange a Pruzbul before the end of Shmittah and one lives in Chutz La’aretz, one may rely on the ruling of Rabbi Moshe Feinstein zt”l (Iggros Moshe, Choshen Mishpot Vol.2 Chapter 15) that in cases of significant financial loss, one may rely on the opinions that nowadays one who lives in Chutz La’aretz is allowed to claim one’s debts from the lender.
Apart from the above, even where Shemittas Kesafim is mandatory, it is still incumbent upon the borrower to insist on paying the money as a gift, and if, upon the lender informing him that he need not repay the loan, he does indeed insist on paying, the lender is permitted to accept the payment as a gift. See Shulchan Aruch, Choshen Mishpot 67:36, and Iggros Moshe loc cit.