Vegetable List.

Vegetables:
Avocado -Small piece
Asparagus Tips
Bean Sprouts
Starter seed sprouts--Broccoli, mustards, turnips.
Alfalfa sprouts
Broccoli tops--Smoothie
Brussel Sprouts--Occasionally
Celery and leaves--rarely and as a side.
Cauliflower--In veg smoothie
Carrots-Shredded or baby ones--also in smoothie
Collard greens--also in smoothie
Cucumbers
Cubanelo Peppers
Red Peppers
Yellow Peppers
Green Peppers
Parsley--Only Occasionally
Spinach--Also in smoothie
Mustard Greens--Mine at times like it--also smoothie
Turnip Greens--smoothie
Kale
Sugar snap peas--love to tear apart and get the pea.
Yellow squash--also smoothie
Green squash
Roma or vine ripened tomatoes--only occasionally
Cherry tomatoes--Same as above
Grape tomatoes--same as above
Mix vegetables--limit corn, also in smoothie
Green beans
Green peas
Corn in moderation.
There are other veggies that are safe for sugar gliders.

Update:
Baby bok choy