Caveat: I have no specific knowledge around ExxonMobil internships…but with that being said.
Most internships are used as recruiting tools by companies. View it as an extended try-out. It allows them to get to better understand the recruit, determine if they want to recruit them, and helps to sell the company to the recruit.
For this reason, many (but not all) engineering internships go to students after their junior year, when they are closer to graduation. Your chances at an internship are much higher after your third year.
You can still get an internship after your sophomore year, and even after your freshman year, but your chances are lower, and your options will be less.
However, you should still make the effort, as going through the process, creating and tuning your resume, doing interviews, doing research on companies, etc., can only make you a stronger applicant the next year.
Also, don’t limit yourself to ExxonMobil. Apply to 50+ companies (not that hard to do online), including smaller ones that may be more willing to offer to a sophomore. An internship with a local company (even if it’s not directly related to your major) helps to build up your resume and makes you more competitive for future internships at much larger companies like ExxonMobil.
A standard Co-Op is a bit different, since it’s usually setup as 3 semesters working with a company (work/school/work/school/work/school), so they may be more willing to take sophomores than a standard internship.
Also look into the possibility of a Fall or Spring internship, these are slightly less competitive than summer internships.
Finally, having gone to a local CC doesn’t hurt your chances as much as being a sophomore. It did limit your ability to build up your resume by taking advantages of the opportunities offered at a university, such as being on a design team, undergraduate research, and leadership positions, that could have helped build up your resume. Now that you’re at UH, you should be looking into these type of activities. 
Good Luck!