Not that Trump wants the Vaccine "only for the US"
The truth, like with anything, is probably in the middle. In that what ever nation developed the vaccine their citizens will be the ones to get access to it first, logic dictates that if the vaccine is develop in Germany it will be distributed around Germany first, then other nations shortly there after (probably EU nations next, then the UK, then the US which could be weeks later). Most likely The American government would want that to happen in the US first.