I keep my mare out 24/7 from march till November and she comes in during the evenings during the winter, more to rest the field than for her benefit.
I don't have a school but am lucky enough to have a large flat field that I can use for my canter work, dressage and show jumping practice. Plus most dressage and sj in eventing are held on grass anyway so I find it really good to practice on grass rather than a surface. I have regular lessons as I find it very beneficial to have somebody shout at me, but I do a majority of my schooling at home on my own or my mum will pop out to give me some help with poles if jumping.
During the winter, I will practice schooling while out hacking. You can do all sorts like medium and collected trot, lateral work, transitions, halts, rein back etc and I practice my canter work and transitions on bridle ways. The only thing you can't really practice are circles and serpentines unless you can get into a large suitable area.
I compete my mare in BS up to 1.15m and BE Novice level and I am aiming her at her first 1* and moving up to intermediate this year. If I have a later time, I will always leave her out the night before a competition as she is so much more chilled that way. So yes, it is entirely possible to compete a horse that is kept at grass.