This is Martha from the Shopify Team.
I think this post would be more suited to the Shopify Design Forum topic, so I am moving it over there.
This change to the Supply theme would likely require some custom coding and reformatting, so, you might find some tips here or by checking in with an Expert.
The Supply theme is not set up to have a cart function that comes in from the right side. The options it has built in Drawer, Page, Flip, Modal and you can preview them in Online Store > Themes > Customize Theme > Cart Page > Cart Type.
These free themes have the function you are after, and might be worth previewing in your shop:
View some more theme options with a slide out cart here, on the Themes page.
Let us know if you have any questions about your shop by emailing firstname.lastname@example.org!
Best of luck :)