If you are talking "organic" as in what kind of chemicals and possible human toxins -- well, I don't know what to tell you.
If you know what chemicals they used, you'll want to research the residual period and the MSDS on each at the very least. You'll also come across extreme websites so try not to get too caught up and stay clinical until you can form an informed opinion. You may then have more persuasive arguments with which to discuss with your boyfriend too.
Let's just ignore that aspect and start with physical damage to the garden plants themselves:
What kind of herbicides are they using? Broadleaf herbicides typically used on lawn will affect vegetables (except corn probably). Is it pelleted? Spray?
There will be danger of wind drift from sprayed herbicides "spot" sprayed often along edges in a long swath.
The roots of the garden plants will definitely commingle in the affected (or "contaminated" if you want to use the stronger word) soil only 18 inches below, not to mention wicking and leaching through the soil if the raised bed is bottomless with no barrier. I think contaminated urban area gardens are made with higher raised beds with barriers or actually raised above ground boxes with bottoms.
I think at the very least, you would want to fence off a buffer area around the raised bed where they are not to treat?
I don't really know what kind of pesticides they use, but most likely they would be broad spectrum and seriously affect the biodiversity of insect populations, beneficial insect availability, and good micro and macro organisms in the soil foodweb.
I actually have a similar issue because my next door neighbor has started using a lawn service. I already don't plant any edibles along the fence line with my opposite neighbor. Last year, I put up a plastic sheeting on my picket fence in an attempt to block drift. But I'm not convinced that that will do any good. Thinking of solid material of some kind, and not planting edibles along a buffer zone on this side too. Luckily, our property is slightly upslope of theirs so runoff is not as much of a concern.
