Yes you can. Grass clippings are a good mulch for the vegetable garden. They help retain moisture, insulate the soil so it doesn't get too warm and provide nutrients as they break down.
I've never had a problem putting them directly around plants right after cutting but many times I'll pile it up near the plants for a day or two if I'm really hot and tired.
I knew someone that used to only compost their lawn clippings. They'd pile it up in a corner of their yard. Never turned it or anything. Broke down fine but others have problems with it.
I try to return as much of the clippings to the lawn as possible though because it cuts down on water and fertilizer needs. Sometimes I'll spread it out, one week I'll bag for the garden, one or two weeks mulch mow.
There can be issues with weed seeds spreading to your garden but if you have a couple of inches of grass over the beds most won't survive.
I'd strongly recommend not letting your grass go that long between cutting though. It can greatly impact the appearance of your lawn. Try to time your cutting so that you don't cut off more than 1/3rd of the blade at each mowing. I know of a few people that don't mow their lawn regularly. Some just once a month, others much less than that. One of them used to look great as the previous owner took really good care of it. Now they all look like crap and spread weed seeds all over the neighborhood. Be a good neighbor and please mow your lawn regularly or at the very least before seed heads develop.