I use winter rye each year and I believe it is essential for the way I garden. I can't speak for hairy vetch.
I throw down the seed about this time, (already down) and in the spring, depending how high it gets, I either mow it and till it in or just till it in. Depending on how much labor you plan to use, you might just be able to turn it over with a spade. But I suspect that 625 square feet might be alot to do. Some have said that cutting it close with a mower is enough, but I feel working it in the soil is a must. It can be a tough grass to kill sometimes.
I think a few on this board will chime in on a no-till method for you.
Good luck!

I also am a believer in winter rye. The years I forget to plant it, I don't get any tomatoes the following summer! True story.

In the spring, I cut down the greens and just leave them to mulch in. And I dig the holes for the tomatoes and plant them. I usually have to cut by the rye a couple of more times and then it dies back by itself. And it has enriched my soil so much.

I'm suspicious of vetch, as I know that crown vetch is very persistent. I don't know about hairy vetch.

Vetch is fine. The secret is to dig or till it in before it flowers.

In fact, that's true for all cover crops.

With cover crops that flower, all get large and/or stiff, thus difficult to mix in. And the legumes have used up the nitrogen stored in their root nodules.

For many years now I have simply gone into the Asparagus patch where the Winter Rye is and stomp on the growing Rye plants so they lay down and become a mulch and the plants die, or turn brown and do not produce a seed head. There is, more often than not, some Vetch seed in that Rye and that too gets stomped on and it too dies when it is stomped on. Now, of course, if you need a clean seed bed that is not they way you should do it.
All cover crops, or green manure crops, should be cut just as they start to develop seeds since that is when the plants have the most nutrients.

Thanks everyone!! I hope I don't have too much trouble with it. But from the sounds of it, even if I have to cut it a few times, it should be more than worth it.

I like planting crimson clover, hairy vetch, Austrian winter peas, and radish/mustard greens as my legume/non-legume cover crop mixtures throughout the fall-winter season.

Since I am a no-till organic farmer, I mow and smother my cover crops 2-4 weeks before planting my crops next season.

Happy Gardening!

If I knew I'd be in the same house next spring I'd plant some of this right now or a month ago. Problem is I still have tomato plants full of fruit. I ripped out half and put them in the compost already, but also have carrots and onions that don't want to finish yet. I also may be moving.

I've shopped around and this mix is the best price I've found for a cover crop. Some people prefer a single variety.

Here is a link that might be useful: soil builder mix