Recently, I had a mishap involving my Firespray, YT-1300 and a puppy that resulted in pegs being broken in many different places, and part of it is broke off into the YT-1300's hole on the bottom of it.
With no funds to spare for replacements, is there any way to still be able to use these? I really don't want to throw them away or let them collect dust.
Super glue?
With the peg broken into the bottom of the YT you might be able to glue the rest of the peg on and just keep it in there all the time.
Sorry to hear about your mishap.
Is the model and base intact.. If so replace the entire peg with a rod..
Corsec do one as system that is really cool http://corseceng.com/omni-stand/
But you could make your own fairly easily, using Brass rod.
Fantasy Flight is pretty great about replacement parts, and from what I've heard people have had a lot of success getting pegs from them.
Never had to try removing a broken peg from a ship (thankfully), but you might be able to use a smallish drill bit - drill into it a bit, and then pull it straight back out. Hopefully the bit will bind, and extract the broken piece.
Had a peg break in a HWK. Used a combo of knife and pin vise to get the tip out. Just be careful not to stab yourself.
Recently, I had a mishap involving my Firespray, YT-1300 and a puppy that resulted in pegs being broken in many different places, and part of it is broke off into the YT-1300's hole on the bottom of it.
With no funds to spare for replacements, is there any way to still be able to use these? I really don't want to throw them away or let them collect dust.
I had an X Wing break at the base so I had a post in tact with connector broke off. I super glued the connector back on with a gel glue (liquid to runny). Problem was that I did not set right as it was 1 day old and I did not know there is only 1 way to put together. It is sideways, but still a beautiful ship that has the card sidways for aesthetics
I did see that pegs and bases can be bought. Check local game stores or specialty shops on Metal Bikini site
If you show a picture we will be able to help even more.
have you tried heating up a needle/pin until it's red hot and melting it into the broken piece?
wait for it to cool down or just pour some water on it and simply, pull it out.. the piece of the peg should come out before the needle pulls out as long as it has cooled down
Email FFG they replaced my missing pegs free of charge. That sorts the missing pegs ... then the actually ship.
I would use executors idea (and have used that in the past with other models) very simple
Yes, this has happened to me with other stuff. Use a tiny paper clip or other wire with a slight bend (like the bank maneuver template). hook when you melt it in and let it cool. Then, when you pull it out, it will take the broken piece with it.
Is it true that FFG will replace pegs? They helped me by replacing some missing parts that came that way, but I have lost pegs due to my own or my opponents' mistake. Will they replace those? I wouldn't even mind if they sold a small bag of 10 for 2 bucks or so. That would be reasonable and help us old men who can't see clear plastic once it hits the shag carpet. 
They have confirmed postage of mine. Awaiting delivery.
No charge has been made.
Here is a tip for removing broken pegs out of bases (I have sadly had to do this twice):
Take a pin/needle/thumbtack/safety pin or something long those lines, and a lighter. Heat the tip of the pin/needle or whatever until it is good and hot (remember metal conducts heat very well so be careful not to burn your fingers - long pins work well, the metal will get hot) then quickly stab the pen into the broken peg (try and get it close to dead center). Press firmly and the hot metal should melt into the plastic. Let the pin cool, then quickly yank it out - the plastic should stay with the pin. If it doesn't work, try a second time and go a little deeper.
Good luck!
What did you use to replace the broken peg?