I don't think the Feds could get an injunction under current laws. In fact, there are several other states which already allow people to secure driver's licenses without regard to the legality of their immigration status.
But, if we wanted to, the Feds could pass some new laws of its own. For example, the Congress could pass a law requiring motorists to secure a new Federal driver's license before driving on public streets and highways. We could have a United States DMV! A lot of folks would oppose that, but I'm confident that the courts would find that Congress has the power to do that if it wishes.
Another thing that the Feds could do is to require that voters in any Federal election possess a Federal ID Card, which the government could print and issue. This would in practice force states and localities to require voters to utilize the Federal ID Card for all voting purposes.
The real problem with illegal immigration, however, is that there simply isn't the will in this country to end it right now. This driver's license law, stupid as it is, really doesn't do much other than to give talk show hosts a way to complain about illegal immigration without advocating steps that are really necessary to end it.
People don't sneak into our country to secure a California driver's license. They come here for employment opportunities that they can't find at home. The only sensible way to stop that immigration is to go after the employment relationship by making it more costly (in the form of penalties) for employers to hire illegal immigrants. But, since doing that would amount to stepping on some very important toes in this country, that isn't being done.
So, instead, we're just going to continue to hear folks call for the construction of immense walls and the posting of unavailable armies at our borders. In the county where we live, illegal immigration has been just another fact of life, at least since I was born a long, long time ago. ;-)