How to ask for a referral without it being awkward: give the person everything they need to say yes in one message

That means the specific role link, two lines on why you fit it, and an explicit out - "no problem at all if you would rather not". What you are asking for is thirty seconds of their time in an internal tool, not a character reference. Most people will do it for a former colleague they remember as competent.

The version that fails is "let me know if you hear of anything", which puts the work of finding you a job onto someone with their own job.