I wouldn't be so sure. The sentence as given actually doesn't shock me.No, in idiomatic English you should a use a subordinate clause with "that": I suggested (to them) that they relax for a while. If no object is mentioned, you could say "I suggested relaxing for a while". So it depends if there is an object or not.
Well, perhaps not, but as you know of course "idiomatic" isn't the same as "grammatical". Correct sentences could still sound weird. Now that you've said that I'm beginning to have a few doubts as to my outright rejection of the original sentence, but in general I stand by what I said.I wouldn't be so sure. The sentence as given actually doesn't shock me.