Vegan Alternative for Pecorino Romano in Pesto
I've used vegan shredded cheese products manufactured by a company named Daiya and have had quite a bit of success. You can find them on Amazon, but products with similar ingredients and labeling might work.

I would recommend a mixture of their parmesan and mozzarella products to get something sort of close to the creamy consistency of pecorino, then add a tiny bit of porcini powder to give it a bit more subtle flavor (a good aged pecorino has a slight tartness to it). 

You'll need to experiment, but I've used their 'cheeses' and they melt / blitz rather well. 

In a pesto where everything is shredded and blitzed, texture is actually less of an issue. I know you could use the above two types to get the taste; you could probably make something that most people wouldn't even think to question. 

But, as with all substitutions, some experimentation will be needed.