Is MD5 particularly vulnerable when given multiple MD5 hashes using a common, secret suffix?
is there a better than brute-force way of finding the value of suffix?  Is there an algorithmic vulnerability of MD5, relying on the internal state that would allow me to do this?


The best attack better than brute-force I could found is described in this paper, it has a complexity of 2^123.4 and a memory complexity of 2^45×11 words. AFAIK this is not practical


  Or, equivalently, to construct a fake prefix that would generate a known hash when concatenated with the unknown suffix?


This is not equivalent to your first question. Your first question is a preimage attack, meaning finding the original value (prefix + suffix) that generated the hash. 

This question can be interpreted in two ways: 


You're asking for an algorithm that can generate a second pre-image, meaning a second value different from the original one that produces the same hash. In this case there is no known second pre-image attack against MD5
You're asking if it's possible to find a collision for a given hash using a fixed part in the message. To this question I only know there are algorithms to efficiently find collisions for a given MD5 but I don't know if those methods can be adapted to use a fixed part in the message (The suffix)