I’d just say “I’d like to put $100 on my metro card.”

I would not use the first one. Technically, it is not wrong. But in terms of usage, it would not be common. The other three statements are fine except I would say 'on my metro card' instead of 'into'.

"Put" sounds the most common to me, but "store", "deposit" or "put" could all be used correctly. If you used "store" you would probably say "on" your card rather than "into" your card. You could also "put" the money "on" the card, although "into" is also fine. I suspect that there are slightly different ways of talking about this in different English speaking countries.

Hi, the first two sound a little strange, but people will still understand you if you said that. The third one, "I would like to put 100 dollars into my metro card?" needs to be changed to "I would like to put 100 dollars on my metro card." "How much do you want to deposit into your metro card?" should be changed to "How much do you want to put on your metro card?"