Yes, it is possible to run autoresponder campaigns in odoo. In odoo, they are called Mass Mailing campaigns and you must install the "Mass Mailing Campaigns" module to get started with it. After install, go to Settings and under the "Configuration" header click "Marketing". Then, activate "Manage Mass Mailing Using Campaign" checkbox, check demo date if you want to see how it may work. Once you install the campaign feature it will redirect you to Mass Mailing and you can set up campaigns.
In the "Settings" area, under the "Configuration" header, there is a menu item called "General Settings. If you click, you will be shown an option on configuring your outgoing e-mail server. It's already configured by default on SAAS. However you can customize it to your own outgoing e-mail servers should you choose. For very large mass campaigns, it is a good idea to have a seperate e-mail server. One that has a proven record of delivering large volume quickly.
These servers provide outbound e-mail services only and you only pay for what you use, making them very cost-effective.
You can do your own research to determine which server is best for your mass mailing campaign. Below are two examples:
Amazon Simple Email Service (Amazon SES)
Mandrill (by MailChimp)
Below is another forum question where a user asked about problems with deliverability with the builtin odoo servers. If you scroll down, you will find that someone suggested using Mandrill.
Here is another question here on the forums that was similar to yours. The opinion there is that odoo's autoresponder system is not as user-friendly as paid services such as mailchimp, which seems to be a common complaint.
Finally, here is a Youtube video outlining the basic steps of setting up Mass mailing campaigns. The video is for an older version of odoo (formerly openerp).
You can create and manage your campaigns by clicking on "Marketing" in the header menu.