For anyone having trouble donating sperm, you need to get to 5 stamina first, easiest way is going into the dining room, there's a shower room that isn't listed in the room overview. Head in there, go into the shower and you'll have the option to jerk for +1 stam.
Various story events also give you some stam, but most of them are locked behind money hence the catch 22. You can work in the bakery or onsen for a little bit of cash, but just using the shower to unlock donating is a lot faster.
If Dev reads this, options that you don't have stam for should not be hidden, they should appear but be greyed out unselectable and say "needs x stamina". The systems in this game are overall really excellent for this type of game, but this is one flaw that needs some fixing.