There isn't anything wrong with drilling through the frame...but...
There's always a "but" isn't there?
You don't want to encourage rust by drilling a hole and leaving it "raw" or allowing an "eye" bolt to wear the paint off around the hole. Go ahead and get yourself a really heavy duty "I" bolt (or two if you want to do both sides) and also get at least 4 heavy washers. Drill your holes and spray outside and inside thoroughly with Rust-oleum or similar rust prevention paint. When you put the "eye" bolts through, make sure there is a washer between both sides and the trailer frame of both bolts (if you do two) so that the bolt will wear the washer and not the paint off the frame. Also, if you go the "I" bolt way, you'll find that the bolt makes a loop and one side is open against the shaft...make sure that points rearward so that if the trailer does come loose, the chain won't pull the eye open enough to come off. I would add a lock washer on the inside, between the nut and the washer and, for cheap insurance, I would "double nut" the eye bolt. Put one nut on and tighten it up, but, not so tight that you start to deform the frame
and then put the other nut down, really tight against the first nut. That will keep the two locked together so that they can't come loose.
To attach the chain to the eye bolts, you'll need some of these:
Chain Quick Link
You can get them at any hardware store and they will make a strong connection between your new chain and the eye bolt (as strong as a welded link).