There seems to be nothing promising on this short of a VERY extensive search, following vague leads in organizations that possibly might be interested in this issue. For example, the following Google searches may find sites that mention useful organizations, far down in some very long lists of hits:

"political dynasties" "U S" site:org gets a whopping 3,000+ hits, but many are books.
("political dynasties" "U S" site:org) not books gets 400+
allintitle: "political dynasties" site:org gets 800+ hits

It might be best to try more specific searches first:

allintitle: "political dynasties" "u s" 30 hits
allintitle: "political dynasties" american 65 hits
"political dynasties" american nonprofit site:org 200+

I found little that seems promising in commercial databases. In Proquest Dissertations the search: anti nepotism gets 2 hits, the first of which is:

Three essays in political economy by Kim, Kwang-ho, Ph.D., University of Pennsylvania, 2004, 113 pages; AAT 3125846
Abstract (Summary) This dissertation consists of three essays in political economy. In Chapter 1, we analyze the inefficiency that may arise in the form of reverse discrimination in the presence of favoritism or nepotism. Favoritism is typically associated with inefficient transfers to the core support of the incumbent government. But inefficiency opposite in nature may also arise through the electoral process in a political environment where favoritism is pervasive. We show that if the policy maker is sufficiently office seeking, a socially efficient action may never be taken if it yields benefits to his core support due to reputational concerns. Hence, the core support of the incumbent may fare worse than other groups. We also consider the implications of policies such as anti-nepotism

Also in Proquest Dissertations, the search: laws and nepotism gets 11 hits, mainly about public schools, businesses, and other countries
The database America: History and Life gets 16 hits for the search: political dynast*. The search: political famil* gets 8 hits, and dynasties gets 172, 144 of them in academic journals. JSTOR gets 98 hits for "political families" AND america*, and may be your best bet for finding research on the topic. But it seems doubtful that many, if any, of these articles were funded by organizations devoted to the study of nepotism, or "political families", etc.