According to Cisco, it is possible to upgrade a 6500 running IOS in native mode without powering down the switch.
However, I have never seen this done in production. The procedure I have seen followed most often is that a supervisor is upgraded off line, and the config pasted in via console. The switch is powered down, both old supervisors are removed, and the switch is booted with the new supervisor. Once it is confirmed to be up and running clean, a second supervisor running the new code version is inserted.
In the document above, Cisco also mention:
Note: This procedure can impact the data traffic.
Cisco recommends that you perform this procedure
during a scheduled maintenance window.
I would suggest that you get an agreed maintenance window, and advise your users that there might be service interruption. This is usually much preferred to advertising no outage, and subsequently causing one. :)