* It's possible that you could tokenize to each provider before making a transaction. But you'd need to see if the provider has a timeout on those tokens if you don't actually make a transaction with it. You'd also need to make sure your processor will even allow you to connect to them directly and authorize a transaction with that token later without consumer interaction.
* You can tokenize the card number and expiration, but not the CVV. Some cards you would be able to get an auth without a CVV, with a slightly higher processing fee (that may or may not get passed on to you). But some will just get declined without the CVV. You are not allowed to store the CVV.
* You're really supposed to get the CVV for any transaction that's an initial transaction. This is a recurring transaction, but since you are changing the agreement (by moving providers and a completely different merchant account), it's really a new initial transaction. Your customer will see a different descriptor on their credit card statement, they may complain, it could end up being a whole thing. And if you're worried about needing to use multiple providers, you don't have your own merchant account, so the bank doesn't come to you, they go to Paypal, or Stripe, etc. Now you have to explain to them why there's a new transaction with no consumer interaction, if they contact you to ask at all before shutting you off.
* Unless you're using a plugin like Stripe's for every processor, quickly doing a request to your processor directly in the user's browser and submitting each of the tokens to yourself, doing this would require you to at least accept and transmit credit cards yourself, not sending the consumer onto the processor's site to fill it out. You are now transmitting credit card data yourself, which means you'll have to go through the PCI Compliance process. It's a self-assessment, and you can skip a lot of steps if you're at least not storing the card number (which you probably should not), but it still a bit of an arduous process. No one's going to check you on this unless there's some sort of breach, but you should still definitely do it.
This is the biggest hurdle to tokenizing at multiple providers by far. As soon as you actually "touch" that credit card number (not Stripe/etc's SDK or iFrame) everything changes. Also the PCI Compliance process is a bitch and a half and is confusing as fuck. For me it was a janky web-app form that was not clear and lead to me answering wrong multiple times then being very confused at the questions it was asking me (I had to call in and their support knew the pitfalls and were able to steer me back but the instructions/question-wording is terrible).
i don't know if their terms allow this to be used for recurring payments, it is used e.g. for services such as steam and humble bundle, where you can choose to remember your payment method.