If the colt is solid, a stallion report filed with PtHA is required which means the OP would need the registration info on the APHA sire (which they don't have since they never got the papers) and pay extra for a late stallion report. Unless the foal being registered is already registered with a recognized association that had a filed stallion report (AHA, AQHA, APHA, etc).
If the colt is pinto patterned and gelded, he can be hardship registered with PtHA without needing to know any information on sire and dam (since the OP is missing all that information), no stallion report required, costs $125, not as good as regular PtHA papers as it has substantially more limits if you want to show at PtHA shows but you would at least have some type of papers.