Does the Navy issue a Pea coat or should I buy one?

It is not issued, if you want one you will have to pay for it. Look at the Thrift Stores first but WAIT!! Until you are stationed at your command. The reason? Because you might spend money on it and then be stationed in a place like Hawaii, Diego Garcia, or Singapore where you will never wear it at all.

The P-Coat is an initial uniform issued item and covered in your clothing allowance. You'll get it at boot camp along with your other uniform items. Personally, I found the "All Weather Coat" (with lining) warmer for standing watches and with service uniforms and the NWU Parka warmer when wearing NWU. I only wore the P-coat a handful of times in 30 years. Manning the rails in the dead of winter is about the only time it is prescribed.

Yes you will be issued one, no none of your uniforms are free, the cost is removed from your pay. You will be wearing your parka most of the time, with the liner, a scarf, a ski mask, gloves, etc. It is much warmer.

You should be issued one along with the rest of your "seabag" full of all of the uniform items you will need. This isn't exactly "free" as it will come out of your first pay check, but you will be reimbursed for most of the costs later on.

Question is would you be allowed to wear it if not an issue uniform overcoat/jacket ?

Buy one by all means but wear only if it's allowed!

You'll get one at great lakes, it's not free.

Yes they're warm and somewhat water resistant.

Mine was issued in 1967. Today?????