I do not have a Bachelor's degree. Would this put me at a disadvantage when getting a visa?

Yes. It will make it difficult, but not impossible for companies to sponsor your work visa even if they want to hire you.

Please visit this website: Immigration Services Agency of Japan to confirm your situation.

There are two exceptions to the college degree rule:

1. If you have 10 years of work experience.
2. Pass one of the approved exams by Japan's Ministry of Justice.

This blog post has good information about getting a work visa in Japan.