Married filing separate, do we both need to download the app separately or is there a way to use one username to do my tax filing then do my wife’s tax filing after that

The tag below your question indicates that you are using TurboTax Online. If that's true, you don't download anything. You sign in to TurboTax in your web browser.

A TurboTax Online username can only be used for one tax return. To file two separate tax returns, you and your wife have to each use a different username.

See the following FAQ for instructions to start another tax return with a new username.

How do I start another return in TurboTax Online?

When you have to file multiple returns, it's usually less expensive to use the CD/Download TurboTax software. You pay for the software once and you can use it for multiple returns. There is an additional charge for each state e-file, but it still usually works out to be less expensive. The CD/Download software also has features that are not available in TurboTax Online. The most significant feature is forms mode, which lets you look at the forms while you are working on the tax return, and even make entries directly on the forms.

Why are you filing as married filing separately? You will almost certainly pay more tax that way than if you filed jointly.