Can I get colored or daily contact lenses when I have amblyopia?

Amblyopia & contacts. Assuming you > age 11 (old enough for cl) you may be able to wear cl with amb. Cl will correct your vision as effectively as glasses but will not get rid of the amb. If your vision in 1 eye due to amb is < 20/40, I would not recommend cl. This is b/c glasses protect your eyes & you have only 1 eye that sees well enough to drive. Also contacts increase risk of vision loss due to infection...Risky.
Yes. Colored lenses are generally for cosmetic reasons and can be worn in each eye irrespective of the need for eyeglasses. Hopefully you can get them to correct the vision in the good eye but the amblyopic eye can get a matching lense even though it might not see well.