Not really just copying the program files won't work in most cases.
If he is using steam he has no problem because he can just install steam on whatever computer and download/install whatever games he has purchased through them.
Replacing a standard dvd drive is super cheap these days. If you have access to a microcenter
To be more specific I would need to know the exact games in question here. or fry's or other large electronics outlet like that you should be able to find something reasonably priced.