The most obvious person to contact to line a box with lead will be a traditional plumber - not the modern type who mess about with PVC pipes, but the old fashioned variety who work with lead pipework, lead guttering, lead flashing and boxings on roofs, etc. The majority of these guys seem to work for the larger roofing contractors (who generally call them "lead men" or "lead dressers") but there are some independents out there.
To DIY you need to be able to cut lead (Stanley knife and straight edge), boss it into shape inside a templet (bossing hammer/mallet, lead dressing sticks or for smaller stuff chase sticks) and finally to solder the corner joints where you are joining pieces (ultra heavy soldering iron or portable gas brazer/welder). Trimming is done with sheet metal shears and lead files and you may also need a chasing-in stick to finih into rebates, etc. Useful advice: wash your hands after handling lead but before drinking or eating (obvious, really, but.......)