How and when can I access my 2020 tax forms with my order number and order date?

You have to log in with the same account.  How to Access prior year online returns
https://ttlc.intuit.com/community/prior-year-return/help/how-do-i-access-my-prior-year-return/00/27010

If you can't get the side menus to open up to access the prior year..... You need to start entering some basic Personal Info in 2021 for the side menu to open up. Just continue a little ways into 2021.

If you used the Desktop CD/Download program then the only copy is on your computer and not saved or stored online. So you need to make and keep your own backups.

Or request a transcript from the IRS
https://www.irs.gov/individuals/get-transcript

It's common to end up with multiple accounts. First LOG OUT of whatever TurboTax account you're logged into right now. Then use this TurboTax account recovery website to get a list of user ID's for an email address. Run the tool against any email addresses you may have used
https://myturbotax.intuit.com/account-recovery/

@ CATHY602  asked:  "How and when can I access my 2020 tax forms with my order number and order date?"

Please explain what that means.

Did you recently purchase the 2020 desktop software from TurboTax and need to download the software to install on your computer?   If so, you can use this link.   Of course, you need to be in the same account in which you purchased the software.

https://shopping.turbotax.intuit.com/unified/downloads/

Otherwise please explain exactly what you need to do.