Can you obtain a CDL with EAD card?

Many trucking company ads state you can.
Might be risky to cross over into Canada or Mexico though. I'd just stay within the lower 48 states.
Do you have a drivers license and if you do, how long have you had it.

 

The EAD allows you to work in the USA without having to apply for a work visa while your green card is being processed, however if you have an employer sponsored green card EB 1, 2, 3 category you can only work for the sponsoring employer after all their the ones that have gone to all the trouble to spend the money to pay for the processing fees with the USCIS. The fact that your asking can you get a CDL licence indicates that this is not the type of work your employer is sponsoring you for. You may even find your EB based green card will be conditional upon you working for them only and until you can apply for those conditions to be removed will have to continue working for them. Once you have received an unconditional green card you'd be free to work for who ever you please. As far as the law goes provided you have a valid SSN you can apply for a CDL .Applying for a CDL and actually working for a transport company are to very different issues as regards conditions applied for an EB green card.