Can you get home tests that tell you if you have adrenal insufficiency or problems with the gland?

No. An endocrinologist needs to be involved in order to diagnose adrenal insufficiency. Usually stimulation testing is necessary. If your pituitary gland is involved then other hormones will also be checked. The adrenal makes lots of different hormones. If the source of insufficiency is because of congenital adrenal hyperplasia or autoimmune disease, testing is a little different.
No. Adrenal insufficiency is diagnosed by lab draw and a test called Cortrosyn (acth) stimulatory test. These are blood draw and not a home test.
Adrenal insufficienc. The acth stim test can be done. Cortisol, acth, aldosterone, renin, potassium, and sodium are tested before the decision is made whether the acth stimulation test needs to be performed. X-rays or ct of the adrenals may also be done. The best test for adrenal insufficiency of autoimmune origin, is measurement of 21-hydroxylase. I do not believe any of these can be done as home tests though.