Storm chasing in June is usually a great idea. It is the second best month after May in terms of tornado activity. The jet stream is weaker than earlier in the season which has both advantages and disadvantages. Storms tend to be slower which makes chasing easier and nicer. The disadvantage is that this could also remove the last “push” the storms need to go tornadic. Moisture is sometimes a problem which means higher storm bases and thus less chance for tornadoes. The lack of moisture can sometimes be “solved” by the fact that you are chasing in higher terrain which needs less moisture.
Chasing in June usually means chasing in the Northern Plains and states like Nebraska and Colorado but there is still a possibility the best chances for storms are as far down as Oklahoma and Texas Panhandle. This means you could be chasing in Wyoming or Iowa one day and miss out on next days storms in Texas. The storms are further apart than in May which could make forecasting and choosing target area difficult.
A benefit of chasing in June is that you are likely to chase in very pretty parts of the Plains and storms and tornadoes tend to be prettier and more isolated than earlier on in the season.
All, or at least the great majority, of the tour companies offer tours in June. You can see all tours in June here.