What version of FileMaker are you using?
Assuming it's a fairly recent version, you can capture the date from table A in a variable and use it with Replace Field contents to put it in every record where the date field is blank in Table B.
Set Variable [$Date ; Value: TableA::DateField ]
go to layout [table B]
Enter FInd Mode 
Set Field [TableB::DateField ; "="]//find all records in TableB where DateField is empty
Set Error capture [on]
Replace Field Contents [No dialog ; TableB::DateField ; $Date]
The above script should not be run at any time where there is the slightest chance that another user might be modifying one of the records in Table B that have an empty date field or record locking may keep a record from being updated.
Hi Phil, thanks for that. It's FM 11 Pro Adv.
Will your script not simply copy across the same date for all records in Table B?
Yes, that's what I understood this text to mean:
"... this date is added to Database A. We would like to run a script from database B that searches Database B for all records where this date field is empty. The script will then link back Database A (where the date will exist) and put the date into the relevant field on Database B."
If that's not what you had in mind, please elaborate in more detail.
Ahh sorry Phil, I didn't make myself clear.
In table A we have a date field per client record - currently 30,000+/- records.
In table B we have a blank date field per client record - currently 30,000+/- records.
The two tables are related by client ref and this is unique to each record.
Each client will have sent their contract in on (potentially) different dates, therefore, each record in table A will have its own seperate date.
We then need to transfer this date to the related record's date field in table B. But we could do with scripting it so that it does it for all the 30,000ish records automatically.
Then in the replace field contents step, use TableA::DateField in place of the variable.
It's a wise precaution to save a copy of your files before doing this so that you can toss out the file and try again if you make a mistake and get results you did not expect.
Phil, thanks. Will give it a try and report back.