How large are the leaflets?

Could be Amphicarpaea bracteata. If so probably not worth transplanting since it is usually an annual.

The leaflets are very small. Too big to be clover but too small to be poison ivy.

Is it native or invasive? Does it flower?

I'd venture to say it's a wild legume, one of the beans or stick tights most likely.

Lycopus probably has it right. Amphicarpaea bracteata has lateral leaflets that are non-symmetrical around the central vein, as shown in your photo.