Does a US citizen need a UK visa for an internship?

The list of activities you are allowed to do on a Standard Visitor visa (or a visa free passport, such as in the case of US citizens) is available in detail on the Gov.UK website. The only reasonably fitting activity would be:

23 Employees of an overseas company or organisation may receive training from a UK based company or organisation in work practices and techniques which are required for the visitor’s employment overseas and not available in their home country.

However assuming you are not an employee of the US branch of the UK company, you are not allowed to participate in an internship as a regular visitor. In this case you should apply for a work visa. The exact type of the work visa must be clarified with your employer, e.g. you might be eligible for a Tier 5 (Temporary Worker - Government Authorised Exchange) visa.

You might be tempted to ignore the advice above and attempt to enter visa-free, lying about the true goals of your visit to the immigration officer. However be aware that such an approach is highly unrecommended and may end up with you banned from visiting the UK for many years to come.

Using the Gov.UK portal, and with my legal head on, I suggest you read If you want to work in the UK in that link. Your internship would seem to be analogous to "work experience", in that its not permanent, is not brief and is likely to be unpaid so I suggest your course of action is to do what it says there Apply for a temporary worker visa if you want to work in the UK for a short time