Can I add live plants to my new aquarium during an ammonia spike?

We need to know how large your tank is, is it a fish in cycle? what kind of fish?

without that information , there is no good way to answer your question.  Live plants will use a small amount of the ammonia,  live plants will help oxyginate your tank wich will be lacking Oxygen during a cycle.  

But you should focus more on water changes, and add prime to the entire tank 2-3 times a week. 

You can add pretty much as many plants as you want, as long as the fish can move and reach the surface, as long as you can still gravel vac, as long as you are going to prune brown leaves and not let them decay and make things worse. could add a large amount of hornwart and just let it float, this way you can still gravel vacuum without worry.  You could add as many plants as you want, but the tank is going to have ammonia untill the bacteria develope and turn it into nitrite, then you will have nitrite until more bacteria come and turn it into nitrate.  

Plants do eat ammonia (and nitrites and nitrates) and help keep levels down. 

If there are fish in this tank you need to do a water change immediately. 

when i got plants i chose some the fish liked to eat