GoDaddy Compatability
I would say that it will probably be ok, but I wouldn't add it to your existing public-facing site. I would setup a subdomain for it and not install any other plugins, allow account creation or use it in a public way (except for CRM tasks like event registration, etc.)

Exploits of WordPress via plugins and themes are common, especially with the default permissions in a shared hosting environment like you describe. Keeping good backups is enough mitigation of this for a regular website, but you will have people's private information in this database, so you need to be much more careful.
Personally, I wouldn't be eager to do it, but if you have no other option, it is possible, just keep it separate from your main website.