I purchased a package from Costco. Can I sync with the local PC and do it on the app or online/cloud?

Since you have already purchased the product from Costco, it makes more sense to go ahead and use that instead of using the online version, which will cost you more. The 2 products are not interchangeable.

I too purchased Turbo Tax from Costco.  Just look for an email with the download instructions and software key.  It will enable you to load the software locally and run it from your PC as a desktop application.  

Best wishes for a good tax season.