flight mode on mobile

Yes.

So they will know I am searching lawyers to sue them?

Are you using your work's wifi network? Or a third party network?

My own WiFi at my own house.

Why are you still using your work phone to look up things on the internet then? And yes, your internet history on your phone will be keeping track of your searches until you delete it (if you have the access to do so).

Because my computer crashed and its the only thing I have to look at internet until I can fix it. I have deleted all my history bit can they tap in while I'm online?

I suppose it's possible they have a tracking program on your phone to monitor your internet activity. I'd assume the worst that they do that (especially if they have some stringent rules on what one can use the company phone for - limiting ones personal use for example).