My Mage character (level 30) has joined the Thieves' Guild. She has 100 destruction, 60 conjuration, 70 enchanting, 50 illusion. She approx 40 lock picking. I've built her on purpose with no sneak (she uses muffle) and no pick pocketing.
My question is , will my Mage be ok in the TG if she builds illusion (gets invisibility) and lock picking?
Or is she completely in the wrong guild without sneak and pick pocketing?

She has already annoyed Maven BB when she used her newly acquired fire storm spell and accidentally burnt all the beehives while taking out the mercenaries. It was fun though!

As long as you can utilise Invisibilty and Muffle (the latter of which you can with your current Illusion skill), you'll be laughing. All my Mages in all TES games have leant towards stealth to aid their magic abilities, so you should be fine.

Sadly, Skyrim doesn't require you to be skilled in any of the Guilds respective skills. You can complete the Companions as a pure Mage, or the College of Winterhold as a Warrior, for example, without difficulty.

But if you're looking for an RP opportunity, look into levelling your Illusion skill to at least 75 and your sneak skill to anything above 40. This way you'll be able to carry on with the idea that your Mage is willing to steal

from/kill anyone who is in his/her way when it comes to aquiring knowledge.

Like any Telvanni would tell you