You're subject to all of PA's transportation laws. I found a good article here:
https://blog.princelaw.com/2008/9/12...out-a-license/
For going to VT like you mentioned it looks like you'd be covered under #14. You'd have to look up and follow the federal FOPA interstate transportation law. Then you'd have to look up VT laws, but it looks like you'd be ok doing what you mentioned. Also now that I'm thinking about it, I wonder if any states issue non-resident permits to people under 21 - you could look that up too and it could get you some other reciprocal states - but probably not PA if I had to guess.
www.handgunlaw.us is a good resource for individual state laws.
Ok now for the practicality: is it a good idea to do all this? Any cop will probably automatically view you as "kid carrying gun illegally" not "young adult responsibly carrying". So you're at a disadvantage from the get-go. Add in the fact you'll have an unserialized handgun. Even though everything is legal, you could still wind up in hot water. You have determine if it's worth it. Of course I'm a proponent of people carrying to protect themselves and others and for recreation, but the fact is that you probably have a higher chance getting in trouble with a gun in this situation than you do needing it to save your life. Do you or your parents have the means for a good defense attorney if needed?
You're also traveling through 'high-risk' area of anti-gun NY to get to VT. Make sure you're an absolute wizard with the applicable laws. Print everything out to carry with you, including the Polymer 80 ATF determination letter.
I'd say - when in doubt, go the conversative route and don't carry, etc. The next few years aren't worth losing your rights to a felony for a lifetime. Even being 21 you'll face the same situations with traveling. When I travel and bring my gun it's stressful to learn and know the laws of the state I'm in. A misunderstanding could put in jail - I'm not a lawyer. Remember you'll almost never be able to open carry in PA in today's environment that there is a constant state of emergency, and you live in Philly. So having it for home defense and fun at the range will be most of your use.
What is your living situation right now? Do you have roommates? First thing I would do is not tell any of your friends about this. Don't do anything stupid (like i mentioned) like taking it out at a party, bragging to someone, etc. Next thing you know it gets stolen because someone talked to someone who talked to someone. Get a very good lockbox or safe - invest in that first. Don't ever be a tough guy. Make sure you know PA deadly force laws.
Do you have a PA driver's license and are actually a resident? Not just living here as an NJ resident?
For actually building a P80 know that they need some fine detail work sometimes and usually a 200 round break in period. But it's certainly a fun project if you're a bit handy. If you mess it up and it only works some of the time - just get another frame since they're cheap enough and make another. Think about getting a custom serial number of your choice engraved to avoid potential cops saying "wtf is this".
Also with all this said - why don't you just purchase a rifle or shotgun with a bit less risk than all this? Not knocking it - just curious why you wouldn't want to go to a dealer and buy an AR for defense/range use in a much easier fashion.
Good luck!