ponerse can sometimes be used as the equivalent for the English phrase
to become in the sense of
I became angry -->
me puse enojado but can it be used in the sense of
I want to to become a [name of career] -->
quiero ponerse un [nombre de carrera]?
I think it's closer to think of it as equivalent to "getting" rather than "becoming".
You can say "Me pongo triste/contento", meaning "I get sad/happy". All uses I can think of it right now are related to emotional state.
but can it be used in the sense of I want to to become a [name of career]
No, you'd never say "Me quiero poner médico" with that meaning, which translates in some possible context as "I want to get medic/medical".
There's also another use for the verb, I hope it's not too confusing to mention it too:
When you say "Me pongo en eso", meaning you are going to work on something (as in "I get to it").
Being not a linguist but just a speaker, I can't give much more precision.
Poner is pretty complex to use. We normally use it when using get.
I want to become... can be translated as quiero llegar a ser..., or quiero convertirme... but the use of poner it's just wrong as follows:
I want to become a Lawyer: quiero ponerme un abogado.
That's like saying I want to put a Lawyer myself.
So it's good to express emotions or adding adjectives to oneself like I become smart (me pongo inteligente), I became disgusting (me puse desagradable), etc.