Managing Security for a Log Shipped db across two (unconnected) domains
but we have no access at all to their domain.


Your Windows user won't work - since you don't have access to the domain or the domains are not trusted.

What you can do?

Ask the vendor to create a SQL Server user with enough rights. Assign appropriate roles, then take that SID and create a login on the secondary server. Remember that log shipping works on the database level!

e.g. Below is pseudocode that will help you:

use LS_PRIMARY
go
create user LS_User without LOGIN;
go
--- give permissions e.g. data reader ,etc

-- get the SID 
SELECT sid from sys.database_principals
where type ='S' and name = 'LS_User'

--- on the secondary server (your side)

use master 
go
create login LS_User with PASSWORD = 'yourSecurePassword', 
SID = 0x125896541236980KIN -- This is the SID from above query