Admiral Tees are screen printers. They probably normally do bulk orders but it might be worth checking whether they've got stock designs (either pre-printed or which can be printed on a 'one-off' basis):
76 High Street, Lowestoft, 01502 511132.
http://www.admiraltees.com/garment.html
It might also be worth trying some of the little shops near to the railway station (e.g. on Bevan Street East)
Norwich Screen Art (15 St Augustine's Street, 01603 630530) are also screen printers. They mention being able to do one-off jobs but it's unclear as to whether they hold suitable stock images:
http://www.norwichscreenart.co.uk/index.php
You might also find something among the quirky little shops on Magdalen Street (on the way out to Anglia Square) or possibly on Norwich Market.
Sorry, I can't think of anywhere in Beccles.
When you're visiting Brighton though, don't miss the wonderful (and very individual) clothing shops around North Laine.
Chris