It's theoretically possible but in reality it's impossible unless you have a carer/PT with you.
Your main problem is, how are you going to load the bars with weights when you need to use both your arms for crutches? You can't standing lift dumbells from the rack as that'll add more weightbearing on your nails, and if you want to lift them sat down you have the same issue as how you get the dumbbells over to your seated position anyway.
Pull ups/chin ups are totally out as you can't be 'crashing' back to the floor on the nails so no commercial gym pull up bar will be suitable as it'll be too high up. Lat pulldowns, well the issue is you need to use your arms to lower yourself onto the bench, but you also need your arms to grab the pulldown bar as you descent - so in reality that's impossible too.
Long story short: unless you hire someone to load the weights for you and bring dumbbells to your seating area, you won't be doing any weightlifting whatsoever unless you can bring your wheelchair to the gym (because then you can carry the plates on your lap, although personally I still wouldn't bother). Doing anything on crutches is totally out and the amount of effort it takes to get a wheelchair to a commercial gym...yeah trust me, you won't be doing it, no matter how 'driven' and a special snowflake with massive willpower you think you are, it just simply won't be happening.