Looking for advice on storing user personalization info to pass to woocommerce

I am newbie and can use advice. I am building site to sell dog training classes, using GP Prime it's working pretty good in test. Now I am working on gathering user info about their dog(s), need dog_name, vet, shot1_date, shot2_date. I have done clunky version, gets the info for 1 dog (4 fields) on the Woocommerce My Account details page and updates to user meta data.

How can I get this info to appear on the single product order page and pass to Woo so that it flows through with all the order data?

I would like to eventually allow for multiple dogs and have a drop down on the order screen to select your pet, but that's too hard for me right now! :)

I know my question isn't specific to GP, but any pointers would really be appreciated.