I'm not a lawyer but I can't imagine this would be problematic from a legal standpoint. For example, I've known several people who go by nicknames or their middle name and their birthday cards always get delivered without issue.

The tricky part is that some mailmen (mailpeople? I'm having a brainfart on the gender inclusive term.) won't deliver mail unless it matches the name on the mailbox. This usually pops up in apartment complexes. In that case, just keep your last name on your mailbox and ensure your mail has a matching last name. Your first name shouldn't matter.