Do we need to submit proof of medical bills being paid with 401K and HSA withdrawals?

You only need to submit proof if the IRS requests it at some point in the future.  You don't need to send receipts with your tax return.

After you input the distributions from forms 1099-R and 1099-SA, TurboTax will ask you if you paid qualified medical expenses with those funds.  TurboTax will then make the calculations (including any penalty or waiver) of the taxable amounts.

You'll want to save any receipts (or attempt to gather what you can) in case the IRS asks you to support those expenses.

And keep it mind:  medical expenses paid from an HSA account cannot be claimed as an itemized deduction on Schedule A. But if you use a 401(k) distribution to pay for qualified medical expenses, you can claim those expenses on Schedule A, since you're paying tax on that distribution (you're just not paying an early distribution penalty on all of it).