Can I install TurboTax deluxe federal and state that I bought online at onto my iPad? Alternatively, can I use the TurboTax app and enter my product key to validate it?

Desktop download software cannot be used on an iPad; it can only be used on a full PC or Mac, not on any kind of mobile device.    You can use online TurboTax software on an iPad or other mobile devices.    Online software is priced very differently from desktop software; you cannot use the license key to pay for online software.

Sorry, but no and no.   Desktop TurboTax can only be installed on a PC or Mac, and is not interchangeable with Online TurboTax.   You should be able to get a refund, however, if you can't use it.   If the place you bought it from won't refund it, then you can use this refund webpage if you bought it from TurboTax or an authorized retailer:

FAQ:  How do I request a refund for my TurboTax product?

https://ttlc.intuit.com/turbotax-support/en-us/help-article/intuit-refunds/request-refund-turbotax-product/L834M6uyW_US_en_US