It depends on the person and their injury. A physician would always be a good idea to start with, but other options would be to either eliminate pushups for the moment and see if the pain goes away or start using other equipment (like the ones you have mentioned in your question) so it can take some pressure off their wrists. Elevated push-ups could be another option for your client or even the usage of bands and/or TRX to add some chest presses. Once they get stronger doing push-ups on an elevated surface (bench, steps, boxes, etc.) then you can start moving them lower to the ground until they can do them without any discomfort and/or pain.
I like a lot of these previous posts- modifying the movement, myofascial release, wraps, etc. are sometimes helpful in preventing wrist pain and if it persists the client should consult a health professional.
One thing I will add that helped me with several clients is to make sure the client’s palm is in the proper position. Their weight should be distributed evenly through all “4 corners” of the palm. In my experience women (and some men) often lift the portion of their palm directly below their pointer finger and/or at the base of the thumb. This increases strain on their wrist joint as well as their forearms.
Another pushup modification I’ve used (if wrist strain is the only issue) is having the client do the pushup on a Bosu (on the floor) or a stability ball (against the wall). With either piece of equipment they can turn their fingertips outward and reduce wrist strain. It’s also a softer surface on which to rest their hands, which sometimes helps. They can still modify a full-body pushup to their knees or put the ball/Bosu higher on the wall to reduce the percentage of their body weight being used for the exercise.
The push up is a primitive movement that should be incorporated within most programs. Try to avoid eliminating the exercise out of any program. Remember, in order to people to feel better and move better we must train movements not muscles. The push up is just a pattern that can be fixed through better motor control. Cue the client to grip the ground with their hand. This will create more torque in the shoulder and also put the shoulder in more stable optimal position.
Id disagree with some above posts. Body weight exercises are king in perfecting someone- it gives them the required strength for their weight and build.
If their wrists are hurting, its called a limiting factor. If you want to lift more, you have to take it out.
A good example is deadlifts- a persons grip gives out long before their back is strained. Hence strongmen and powerlifters use wrist wraps.
For a pushup you have two options.
1- wrap their wrists, any athletic tape will do.
2- lighten the load (weight) that their body is resisting. When they are in a pushup neutral position, stand over them. Your feet should have their lower back or waste between them. Link a band under their chest and hold the ends. You then can control how much weight to take off their wrists.