How do you secure your prior purchases in your trailer when you go into another store to make more purchases?
It's hard to secure a bike trailer in a meaningful way.
First, you've got to lock the trailer to something, because it's easy to roll away with a loaded bicycle trailer. That means a secure point on the trailer to lock to, since the tow arm is usually detachable. You also will need a decent lock. Secondly, unless your attach a solid (example: metal) lockable box or cage to the trailer and put your things in that, they can be stolen by a thief with a knife. So, if you want to leave something valuable (like a laptop) in your trailer while you're shopping, this is the only way to go.
But in my experience, thieves steal what looks easy to steal. If you're leaving groceries, merchandise, helmets, etc. in a parking lot for a few hours, just fit a tarpaulin or other cover so that you have to pull back the cover to see what's inside. This usually a sufficient deterrent to someone stealing your beer, and also is a good first barrier to keeping the wind and rain away from your stuff when you're riding.
Unlike in a comment above, I'm not going to assume that you can carry the rest of your shopping when you go in the second shop. But hopefully you can carry any valuable items, then leave cheap bulky stuff in the trailer.
I do this with panniers and backpack, buying the more stealable items first and keeping them with me, then buying fruit and veg, leaving them in an open pannier, then going to the supermarket for the rest of the food shopping. The open pannier is on the basis that a closed pannier could be full of something more tempting than a bag of onions.
You could make your own cover from plywood and some hardware store parts. Or get some plastic storage tubs that have a pass-through for a padlock and bolt those down in your storage unit.
But it’d still be pretty insecure. Most things are just there to keep honest people honest.
Another option is to bike with your vicious looking dog (or child) in the trailer and leave him with the bike. Not sure if that’s a good solution.
My bike trailer was a child carrier, and the boom can retract underneath. I can fit a handle and have a third wheel that goes on the front. This would allow me to push my trailer around the shop/store like it was a pram.
However this is a very rare occurrence, I rarely buy enough to warrant this kind of transport.
Some stores may have a policy preventing the use of outside trolleys - supermarkets might restrict access to prams/strollers for loss-prevention reasons.
Something vaguely similar but not the same.