Barring laws restricting what you can and cannot put in the yard, there won't be much you can do but respond. If there are laws, report them anonymously.

On the up side, hay itself is not harmful. The seeds from it are annoying, but the resulting plants tend to be non-noxious and not an issue in the long haul.

If your lawn is thick and healthy, hay can be mowed straight into it. It'll filter down and add a little bit of organic material to the soil. That's a good thing.

Thinner, patchier lawns can end up with severe weed issues from this. Fortunately, most weeds are annuals and won't survive the first frost (if you live in an area with frost).

Pre-emergent application in your beds might stop or at least slow down the sprout of weeds, but it would depend on what you grow in your beds in the first place. Some plants tolerate pre-M applications very well, others don't, and that varies from pre-M to pre-M.

I use Barricade through the gardens to slow down the sedge infestation (and treat that with Tenacity), and it works very well. Most of my summer plants are tender or semi-hardy and play well with the Barricade (as the end of season two foot deep roots show).

Plant an evergreen hedge of Arborvitaes or the Spruces that deer don't like, along the fence line.
It may not too expensive to buy a bunch of 3' plants for an instant screen. It can also be cheaper at $1/tree when buying 1 foot tall bareroots in bulk..

What ever size you get just be sure they are in a good fertile soil with adequate drainage and retention, then keep them irrigated all season long.

Technys especially really grow quickly with lots of water, but deer will open up the bottom 4' of leaves.

There are 2 or 3 precepts to keeping weeds out of a lawn.

If your lawn is dense and tall, you should have very few weeds.
If you water your lawn deeply and infrequently you should have very few weeds.
If you fertilize regularly (3x per year) you should have few weeds.

If you do those things and still get the occasional weed, then the occasional spot spray with weed-b-gone should handle most weeds.