I’m not sure on the biking aspect, but have had tendonitis before and golfers and tennis elbow and it is usually caused by ergonomics, yes – a combination of posture and awkward angle usage.
It would be an idea to rest the area until it’s improved – I’d suggest using ice packs as well. Physios vary – some are good with gentle stretch exercises some don’t have a clue. So maybe get a recommendation? But from my experience first thing is to stop doing what caused it and rest – it might wear off in a few days but come back as soon as you do the same thing. It might take a couple of weeks.
But icing can help a lot. Gel ice packs for the freezer (or a bag of frozen peas in the meantime). The gel ice packs are good as you can fasten them on with velcro and get them in the right place.
Helps speed up the healing process in the first few days.
Then it’s looking at the ergonomics so it doesn’t come straight back again. IME it’s to do with reach for the fingers as well as angle of the wrist. So maybe adjust lever reach and height of handlebars.
It can start with trapping in the wrist area (carpal tunnel) which goes down to the fingers and up to the elbow. Or sometimes it can start at the neck area if the neck is at a bad angle and work down the arm. But all posture related.
Hope that helps a bit. You could try working through it but it could get worse – then takes longer to get better.