Replication fails with "Transaction log for database is full due to LOG_BACKUP"
there is no clear indication of what may be causing this upsetting in the transaction log.

but I would check it out what is happening in the distribution.

run the following procedure:

sp_helpdistributor


that will show you where the distribution database and server is.

Assuming the name of the distribution database is distribution:

 SELECT * FROM distribution.dbo.MSrepl_Commands 


have a look at what you get and use the last transaction to see what is in it:



check it out:

 SELECT * FROM distribution.dbo.MSrepl_Commands 
--- commands for all transactions for replication for seqno or all
  WHERE   xact_seqno = 0x000002470000F7D30021


if all good or at least nothing obviously bad:
use that xact_seqno here (I did not use the end one on this example - but if your replication is very busy you can limit it):

    exec distribution.dbo.sp_browsereplcmds @xact_seqno_start =  '0x000002470000F7D30021',  -- returns all/seqno info (sql query) on replication
   -- @xact_seqno_end  =  '0x00001C030000F4540018'


This allows you to see what has been replicated - what commands to each article (see on the right hand side of the picture below)



This is not a comprehensive answer but would give you a start.
Well after the obvious thing as looking at the replication monitor.

Check these scripts to give you some extra vision.

You can also check Some useful commands and sp for Transaction Replication