It is theoretically feasible, but in practice, it is unattainable unless you have a caregiver/physical therapist accompanying you.
Your primary issue lies in how you will load the bars with weights when both your arms are needed for crutches. Trying to lift dumbbells from the rack while standing will exert more pressure on your nails, and if you opt to lift them while seated, you face the challenge of transferring the dumbbells to your seated position.
Pull-ups/chin-ups are completely ruled out since you cannot risk ‘crashing’ back to the ground on your nails, making any standard gym pull-up bar too high. As for lat pulldowns, the dilemma arises as you require your arms to lower yourself onto the bench and also to grasp the pulldown bar as you descend – rendering it practically impossible.
In essence, unless you enlist someone to assist you in loading weights and transferring dumbbells to your seated location, engaging in any weightlifting activities is unachievable. Moreover, performing exercises on crutches is entirely unfeasible, and the exertion involved in bringing a wheelchair to a commercial gym is considerable. Rest assured, regardless of how determined or motivated you may consider yourself to be, the practicality of accomplishing this task is non-existent.
