Which biochem company could I apply to for a summer internship?

Where are you located? Any of the companies listed on Wikipedia under the list of pharmaceutical companies and any of the companies making consumer goods. Take a look at who makes the products in your bathroom (e.g., toliet paper, dixie cups, hand soap). Also, I have not met the "requirements" for any of the jobs to which I applied and for which I was hired. I applied anyway. The reason I got hired was because there is no one person who meets all the job requirements. Most job requirements are unrealistic and usually applicants only satisfy a few of the "requirements".

Here is a list of top employees currently advertising jobs on various job boards - I have seen internships on some of these website, but I cannot remember which ones.

Raytheon, Northrop Grueman, Mantech, Lilly, AstraZeneca, GSK, Pfizer, Roche, Honeywell, Kuehne Nagel, Braun, Shire Pharamceuticals, Abbot, Hillshire brands, Mantech, General Dynamics, OCLC, Battelle, Cameron Craig Group, MedNax, Cardiovascular Systems, Cincinnati Bell, Experis, Koch, Arthrex