In your example sentence, "This is the thing I am looking for information on" would be grammatically correct...and preferable to anything else I can think of.I know we can say: this is the thing I need
or: this is the thing I am talking about = thing about which I am talking
but how to say something with a nonseparable phrasal verb?
this is the thing I am looking (for ??) information on
As far as unseparable phrasal verbs are concerned, there is normally nothing particularly wrong with ending a sentence with one, if that's what you're worried about. For example:
This is a policy I cannot put up with.
This is behavior I frown upon.
This is a problem I have puzzled over.
Don't worry about ending sentences with prepositions in this case: in phrasal verbs, the preposition is no longer functioning as a preposition in the usual sense. (In fact, some books prefer to use the term particle.) I actually feel a bit unsure if I'm getting the point of your question. Am I in the ballpark?
Student or Learner