My 4 year old knows that he can never ride his bike without a helmet. There are two reasons for this:
1) Even with training wheels a child can fall, or be involved in an accident. Why take any risk?
2)Chinuch! Do you tell your young children to make Brachos, not touch muktzeh, etc? Even though they have no chiyuv? I certainly do, with the intention of “training” them to be in the habit of doing things correctly. Assuming the younger kids are old enough to understand (and if they’re riding bikes with training wheels they probably are), don’t you want them to start getting the message that we only ride a bike with a helmet on?
As far as your older son, it’s already been said — any bike store should be able to attach a basket for a reasonable cost. Think of it as the same way you would spend some money for any other Mitzvah — the Mitzvah of remaining safe is also worth money. (And it probably costs less than matzos :)!)