A word of caution: pure ammonia, in either its gaseous or liquified form is extremely dangerous to handle, especially in the context of your 'hobby' where, in your own words you "don't have access to a ton of glassware or specialty equipment." Any ammonia that escapes from your reaction and isn't ventilated properly will cause severe burns to anything it comes into contact with, including completely trashing the lining of your nose and lungs etc. In an attempt to relieve your curiosity, a photograph of a birch reduction showing the characteristic blue is shown below:
That said, the question is an interesting one in the context of laboratory chemistry.
Is there a way to use a gaseous form of ammonia for the birch reduction that doesn't generate sodium amide?
Yes and No. The most common way of performing these reactions on a small scale in a lab context is to essentially pass the gas over your reaction as illustrated below:
Practically, you're not actually making liquid ammonia and then transferring it to your reaction, but rather your passing it over your reaction flask which is fitted with a condenser which will cause in-situ liquefaction of the gaseous ammonia. Even a simple dry ice condenser is sufficient to do this, and the formation of the active reagent is easily observed by the appearance of the characteristic blue/green colour. Any gas not liquified is then passed to the back of the fume cupboard, or, ideally, bubbled through a solution of something to neutralise it and hence stop it being released.