Your proof of payment is your bank account statement showing the payment to the IRS. Or if you used the IRS payment website you would have received confirmation numbers for each payment.
No, the IRS will not send you a statement. Your proof of payment is your own bank or credit card records.
You can check your online account with the IRS:
https://www.irs.gov/payments/your-online-account
Did you enter the estimated payments you made into your tax return? Go to Federal>Deductions and Credits>Estimates and Other Taxes Paid>Other Income Taxes
The IRS will not send a statement showing your estimated taxes paid.
You can prove that you paid, with copies of your paid checks or your bank debits.
You can also see your estimated tax payments in your tax transcript.
Please read this IRS page on how to get your tax transcript.