Hi All. Hopefully we've got a bit better at describing these problems than some of the "old school" and really unpleasant sounding descriptions above!
Firstly, it's important to check it really is tennis elbow (healthcare professionals call it lateral epicondylitis or more frequently now Lateral Elbow Tendinopathy (LET)). LET is where the muscles that help with gripping, extending the wrist and turning the forearm all converge onto one tendon (the bit that joins a muscle to a bone) which gets asked to do more work than it has capacity for at that time and becomes irritated. It can be a sudden overwhelming event or a combination of smaller things. Other conditions can bring on elbow pain such as referred pain from your neck and shoulder, so it's worth getting a check over and making sure nothing else is going on.
LET pain can come on from daft things like holding your wrist up on a mouse, gripping dog leads on a long walk, pull-ups, wringing out cloths or as in this case, putting up a rifle. Or more likely, it can come from a combination of many things that serve together to overwhelm the tendon tissue. I'm seeing a lot of these after people have been COVID isolated or doing more working from home on unsuitable tech/table combos. 2 weeks not doing much in the house allows us to lose some fitness and strength so what would have been normal activity is then a big step-up. After the first lockdown, many people haven't regained the fitness they lost with the enforced lack of activity. Kitchen tables are higher than desks and chairs lower than work chairs, so hands on a laptop can then be in a position that makes these muscles have to work differently.
If it's a new pain, then a short period of relative rest (doesn't mean sitting on the sofa watching Netflix all day) can settle the symptoms then a slow build up of the activity again. Perhaps think about the analogy of the difference between running for a bus and running a marathon - both activities are running, but if you have only done the first then the second is likely to make you very sore. In reality, we would aim to slowly progress the running until we were fit enough. The same goes with gripping/twisting/lifting movements in the case of LET - building that fitness in a progressive manner.
Often, the pain will go away all on its own, if you minimise the aggravating activities and build back into them slowly. Sometimes you might need to get some help.
"Physio" means many different things to different people. Generally, a slow build up of work in the muscles that pull on that tendon is the most evidence-based. Other things mentioned here tend to be adjuncts and can help with pain relief whilst the hard work is being done. I haven't ever had to send anyone to surgery (including myself: just because you're supposed to know what you're doing doesn't mean you're not a dick sometimes

). I rarely use CSIs these days but, if the strengthening work doesn't help, ESWT (extracorporeal shock wave therapy - a machine that puts pulses through the tendon) is the next level treatment and can have good results. All these treatments still generally require some muscle work afterwards otherwise the problem is likely to recur.
It's a field that is changing and progressing all the time, so if you're getting help, do ask your healthcare provider how up to date they are. Here's a very recent summary of the current evidence for exercise versus passive treatment with or without invasive treatments to help prompt discussions:
Exercise interventions in lateral elbow tendinopathy have better outcomes than passive interventions, but the effects are small: a systematic review and meta-analysis of 2123 subjects in 30 trials | British Journal of Sports Medicine
Finally - good luck. It's a pain, especially for those of us with jobs that mean you can't just stop the aggravating activity and I feel for all of you having been there.