Allow entering application while in transaction
As I ran tests if I comment out the BEGIN TRANSACTION and just run the sprocs (outside of a transaction) then I no longer have any issue as the tables aren't locked.

Inside a transaction, the modification locks are held for the duration of the entire transaction. Without the BEGIN TRAN, locks are only held until the end of each statement.  There will still be blocking, but it won't last as long, so your site is probably able to operate more effectively.

The customer still wants the site not to lock up and continue this process so I played around with adding:
SET TRANSACTION ISOLATION LEVEL READ UNCOMMITTED;

But this did not help in any way as it still seems users cannot do anything in the application until after the transaction is complete.

Data modification queries don't work that way.  Check out the documentation on isolation levels:

Choosing a transaction isolation level does not affect the locks acquired to protect data modifications. A transaction always gets an exclusive lock on any data it modifies, and holds that lock until the transaction completes, regardless of the isolation level set for that transaction.

What you actually want is the opposite - the nightly process is going to take locks, you want the client (website) to be able to bypass them.  But using READ UNCOMMITTED / NOLOCK is not the ideal way to do that.  There's a better way, which is
Use RCSI
If the web site is mostly reading data, then you might have success with the read committed snapshot isolation level (RCSI).  This would allow the website to continue to read from the database without being blocked by the nightly process.
You can read a detailed account of what to consider when turning on RCSI here: Implementing Snapshot or Read Committed Snapshot Isolation in SQL Server: A Guide