If you live in Seattle, the Mountaineers organization is a great way to go. But, if you do not live in Seattle and will be visiting, their schedule probably not work for you. I strongly urge you to join a guided trip with a approved guide. I can personally vouch for Alpine Ascents international
You will have to cross the unmarked glacier to get to camp Muir. A ice ax and crampons are recommended. Please carry the 10 essentials. The weather can be a killer even in June and July. Please register your hike at the park office.
I really doubt any group will just let you "tag along" to the summit - the liability issues would be huge - If you fall and get hurt you can sue the group.
Solo travel above high camps or anywhere on glaciers is not permitted except with prior written permission from the park Superintendent.
Be safe. Enjoy the climb.
