Can you say this:
Do you want Pizza or we could have burgers. They have that/it here too.
Are both okay?
Your first sentence should actually be two different sentences, due to the different verb tenses. "Do you want pizza? Or we could have burgers...." And while a lot of AmE speakers would blurt it out pretty much as you originally posted it (for example, an exhausted mother trying to please a group of children at a restaurant as quickly and simply as possible): "OK, do you want pizza....or we could have burgers....that have that here, too." But such phrasing is not grammatically correct. If you were to say it to a native speaker, they would understand you, but you would not pass a written exam using that phrasing.
Properly written, the conversation should read: "Do you want pizza? Or would you like a burger? They have both here." Or "We could have burgers - they have those here, too."