The 1099-R is correct and the IRS sending a letter is also normal and to be expected.
If you reported the 1099-R as a rollover with the box 1 amount on the 1040 line 4a with the word ROLLOVER next to it then all you need to do is send the IRS a bank statement form Vanguard that show the same deposit as the rollover amount.
The IRS has no way to connect the receipt of rollover money when the receiving bank is different from the original bank. Any money deposited into an IRA is required to be reported by the receiving bank on a 5498 but that will not tie it to your rollover - you must report that on your tax return. Vanguard is required to issue a 5498 showing the rollover amount in box 2.
See 5498 instructions: https://www.irs.gov/instructions/i1099r#idm140302483335632
Box 2. Rollover Contributions
Enter any rollover contributions (or contributions treated as rollovers) to any IRA received by you during 2020. These contributions may be any of the following.
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A 60-day rollover between Roth IRAs or between other types of IRAs.
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A direct or indirect (within 60 days) rollover from a qualified plan, section 403(b) plan, or governmental section 457(b) plan.
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Any qualified rollover contribution as defined in section 408A(e) from an eligible retirement plan (other than an IRA) to a Roth IRA.
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A military death gratuity.
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An SGLI payment.
For the rollover of property, enter the FMV of the property on the date you receive it. This value may be different from the value of the property on the date it was distributed to the participant.
If this was done in 2019 you should have received the 5498 in May of 2020. You do not report a 5498 at all - it is simply information to you and does not go you your tax return but you can use it as proof of the deposit of the money, but any bank statement that shows the deposit should be enough.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**
