Putting this here for anyone else who has had this issue. Thanks to Daniel Hazelbaker for helping with this!

On Rock v7.3 We've merged people/records before many times without any issue. However, we started getting the following error when trying to merge:

Merge Error
The following error occurred when attempting the merge: Object reference not set to an instance of an object.

Checked out the system exceptions log and it returned the following:

Source: App_Web_axtjchf5

Stack Trace:

at RockWeb.Blocks.Crm.PersonMerge.<>c__DisplayClass27.b__14() in c:\inetpub\Rock\Blocks\Crm\PersonMerge.ascx.cs:line 608
at Rock.Data.DbContext.WrapTransaction(Action action)
at RockWeb.Blocks.Crm.PersonMerge.lbMerge_Click(Object sender, EventArgs e) in c:\inetpub\Rock\Blocks\Crm\PersonMerge.ascx.cs:line 360.

The error indicates that it loaded a Group that should have existed, but for whatever reasons didn't actually exist.
To check if this is actually the case, use the following SQL:

SELECT * FROM [GroupMember] AS GM LEFT JOIN [Group] AS G ON G.Id = GM.GroupId WHERE G.Id IS NULL

For us, this returned about 192 rows of affected data - people that were linked to non-existent group IDs.

To test removing these faulty rows, use the following SQL:

BEGIN TRANSACTION
DELETE GM FROM [GroupMember] AS GM LEFT JOIN [Group] AS G ON G.Id = GM.GroupId WHERE G.Id IS NULL
ROLLBACK TRANSACTION

For us, this also returned 192 rows so I was confident we had a match and were ready to actually remove this faulty data.
To actually remove the data, simply delete the two TRANSACTION lines:

DELETE GM FROM [GroupMember] AS GM LEFT JOIN [Group] AS G ON G.Id = GM.GroupId WHERE G.Id IS NULL

After running the delete statement, we were able to successfully merge people again.