Can my undergraduate research thesis be published in a scientific paper? If so, how should I go?

If your work is good enough, it deserves to be published. It doesn't matter if it is the result of a hobby, undergraduate studies, or your main job as a researcher.

Summarizing it to make it fit into the space allotted in a journal or conference proceedings will be a daunting task, ask your advisor for guidance and help.

As vonbrand has said, it is the quality of work, rather than the degrees behind the person who wrote it, that should determine whether or not something can be published. There are very few journals who have standing bans on publishing undergraduate research, though be aware that preparing a paper for publication is a fairly daunting task.

My advice would be to talk to your advisor about whether or not your work is in the position to be published as a stand-alone paper, or if it's going to be incorporated into a larger paper.

In the first case, the next step is to talk to them about what still needs to be done, what journal to target, and the steps you need to take to get going on that path. In the second case, it's largely a matter of asking if you'll be included as an author on that paper, or end up in the acknowledgements section. Having a clear and direct conversation up from will help things in the long run.

For reference, my undergraduate thesis was published with me as the first author on it.

To add to the above answers, there are in fact, some journals which are dedicated exclusively to the publication of undergraduate research. It might be easier for your paper to get accepted in such journals. The Council of Undergraduate Research provides a list of popular undergraduate journals.

However, you can also submit your paper to any regular journal as very few journals have restrictions on publishing papers by undergraduates. Here is an article on publishing as an undergraduate that you might find interesting:

http://blogs.plos.org/thestudentblog/2014/03/25/final-steps-undergraduate-research-experience-peer-review-publishing/