Debian will mail you confirmation of the bug report, including the assigned bug number. So if you don't have an email after waiting fifteen minutes, something is going wrong. (Although it could be a mistake in the email address you gave to
reportbug, or that some part of the Debian infrastructure is down for maintenance).
In order to have "a mail transport agent (MTA) like Exim, Postfix or SSMTP configured on [your] computer to send mail to the Internet", you would have had to configure it yourself, and also test that it works. So if you're unsure whether you have that... it's fairly clear that you don't.
EDIT: Simply accept the defaults for every option, and reportbug will submit bugs directly to the Debian servers.
Or use the alternative method suggested by GAD3R.
Alternatively, you can configure
reportbug with your internet SMTP server, just like you configured Thunderbird.
reportbug differs in that it does not have a config wizard that knows the values needed for common providers, although they are not too hard to find. However, if you use the most popular email provider, you will have to either "Allow less secure apps to access your account", or switch to two-factor authentication and then generate an "application specific password" for
reportbug to use. The latter approach is very similar to what the "more secure" OAuth does behind the scenes. I use the former approach - it's not a problem if you're somehow sure that the "less secure" apps you use are not storing the password - and reportbug asks me for the password each time I use it.
Do not try to configure an MTA just for
reportbug. Unless you already understood and would remember that this MTA would probably not work for many other purposes! Because running an MTA and getting the world to accept your mail from it is a professional job, requiring periodic maintenance. For example there is an unmaintained series of Ars Technica articles on self-hosting email from 2014; it looks like part 3 of 4 discusses aspects like DKIM.
If you did have such an MTA configured, you might check your local inbox for bounce messages, then escalate to checking the MTA log files.