Can I becomne an English teacher with a Sociology degree?

You can earn your BA in Sociology, then do an MEd (or MAT, depending on how the university delegates a Master's degree in teaching) in English Education.

You have to have an English degree.

You should check your state's board of education site for more information.  You could also consult your dean or counselor at your university.

To teach English, you would need to at least minor in it (and qualify to be a middle/junior high school English teacher) or major in it (and qualify to be a high school English teacher), at least in my state.  I love sociology! and I considered becoming a Sociology teacher too but sociology jobs are extremely rare, unless you want to teach Sociology at the college level.  With a Master's in Sociology, you can teach at your local community college.  

Good luck!

Check with the Dept.  of Ed. In your state and talk to the Certification Div.  You may be able to take a few classes in English to get certification.  You probably only need a few upper level classes. If a district didn't find a teacher with certification in English, you could get temporary certification for a year which would allow you to teach.  Unfortunately, there may be more applicants than positions.  

You should, if possible, take graduate English classes so the classes help you meet your certification requirements and start you toward masters which will give you more money on the salary schedule and meet the 5 year requirement to have masters degree.  However, if you can take classes for certification do that and finish masters after you are working.  Costs could affect whether you
 are hired or not.  With money as it is, even a couple of years on bachelor level rather than masters saves the district money.

Math and science are most sought after teachers.  Reading is another biggie.  

Good luck.