This list is meant to be suggestions. It is not comprehensive. Feeding finches (or any pet bird) is something that needs a little thought and a little preparation. These fruits and vegetables can be chopped small or left in larger chunks.
Apple, no seeds or skin
Banana
Beetroot
Bell peppers
Bok choy
Broccoli
Butternut squash
Cabbage, both savoy and kale
Carrots
Celery leaves
Chickweed
Corn on or off the cob
Dandelion greens
Fennel
Mustard greens
Nasturtium flowers and leaves
Parsley
Peas
Pumpkin
Spinach
Sweet potato
Tomatoes, in small amounts and ripe
Zucchini
This list is as useful for any size of bird, except those with specific dietary considerations. Please also check out Foods you should Never give your Bird.