You can schedule procman to run via Task Scheduler. The challenge is when. If you have a specific time then you can set that. If you don't know when then it becomes harder as you are limited in startup options.
To stop procman from running you can either set a run length limit in Task Scheduler or you can tell procmon to stop collecting after a certain interval or event size limit.
Note however that I don't believe using procmon is the correct solution here. The biggest issue is that you're talking about IIS. The process by itself isn't sufficient to identify the site being run. If you are like most other companies then your IIS is hosting more than 1 site. There will be no easy way using procmon to tell it to monitor just 1 site out of all of them. w3wp is the process IIS uses to run all sites and therefore each app pool is going to have its own instance. Within the app pool any related sites are going to be running. Depending upon your IIS configuration you may be monitoring the wrong process or more than just the site you care about.
Personally I think the better option is to use IIS's custom logging directly, your own custom logging within your app, AppInsights for on premise or the older IIS monitoring tools. These will allow you more fine grained control over the site(s) to monitor and they will better integrate with how web apps actually work.