I am handling membership (about 600 dues-paying members) for a small "shoestring" non-profit using spreadsheets and it is a pain. We have a GoDaddy hosted website (cheapest economy plan) using WordPress. It looks like CiviCRM may be a good choice for us. Will it work if added to our existing GoDaddy environment or will we need a larger, more powerful server? Sorry, I don't have or understand all the technical specifications. :-(
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.