I was in the same boat as you. I got my undergraduate degree in Chemistry and was very skeptical at my chances of getting admitted to a ChemE doctorate program. Department websites state that Physics and Chemistry majors are always welcome to apply, but don't explicitly tell you that you are at a disadvantage. However, there are a good number of people who get admitted to top/decent ChemE departments with a non-engineering undergraduate degree. This is a recent trend that developed from the fact that ChemE research has become extremely interdisciplinary.
Hopefully, you are hoping to make this transition with a pretty decent profile (high GPA, GREs, good letters of rec, etc). Getting into a Top 10 school is possible if you are a STELLAR student. I've talked to a professor in the admissions committee of my school and was told that you can greatly strengthen your application by emphasizing why you are capable and qualified to make the Physics>>>ChemE transition. You can do this by stating why you took or will take certain courses and by fluently connecting your prior research experience to the research taking place in the specific department.
Also, it is not the end of the world to attend a mediocre school (to me, those are in the ranks 25-50). This is especially true if you plan to work in industry. I was thinking about taking a year or two after graduating in order to strengthen my application/resume somehow, but I didn't think it would have been worth it to delay obtaining my PhD.