6000+ members can be run on any free web hosting company, what will matters is how many will be online at a time, how many database queries you'll be handling at any given time, how many concurrent connections to your website, how much bandwidth you'll be using, how much disk space you'll be using, and how many page views you'll be getting, and how big your pages will be (CSS, JS, images, html, etc...).
If you're just trying to estimate...
If your e-mail lists are active and you have hundreds of e-mails a day then I would be conservative and look at a VPS. If you're talking about 100 e-mails a month then any shared hosting should be able to accommodate you. Find a company that has a good upgrade/downgrade path so you'll be able to scale up or down if your estimates are off (ideally a cloud host would work best but it's more costly than a VPS but you can scale easily and it's cheaper and more reliable than a dedicated server).