Yes, Sir Thomas would sit in the House of Commons.

Source: A post from the L&T archives by Julie W:-

http://www.pemberley.com/bin/archives/regarc.pl?read=36644

In the early 19th century parliaments were formed for a maximum term of seven years. A parliament could be, and usually was, dissolved by the King (at the request of the Prime Minister) before the term expired.

UK General Elections were held in 1802, 1806, 1807 and 1812.
If you click on these dates in the list of general elections on the page linked below, you will be able to get an overview of parliamentary politics at the time of MP. Governments were formed by the Tories through most of our period.

Political groupings such as the Whigs and Tories could best be described as "factions" rather than "parties", as the party political system in the modern sense was still in the process of evolving.