Some differences have been observed in certain studies, although most reviews nowadays seem focused on the genetics angle in variations to vaccine response. One such (2015) review cites (somewhat in passing) as evidence for known variation in ethnically linked response a study in Taiwan that found response variation for some vaccines but not for others (the groups involved being aboriginals vs Han Chinese):
children whose parents were both aborigines had lower anti-HBs [hepatitis B] mean titer than did children whose parents were both ethnic Han Chinese. Children of mixed parental origins had intermediate mean titer of anti-HBs. Serologic responses to Japanese encephalitis virus and diphtheria vaccines did not show such correlation with ethnic groups, indicating that the determinant for HBV hyporesponsiveness among the aboriginal children is distinct from that of other childhood vaccines. It was therefore concluded that host factors pertaining to ethnic origin might be responsible for the hyporesponsiveness to HBV vaccine in the aboriginal populations
Such differences in the immediately quantifiable immune response were observed in other studies, e.g., one in the US (on response to rubella vaccine in African-Americans vs others):
individuals of African descent have significantly higher rubella-specific neutralizing antibody levels compared to individuals of European descent and/or Hispanic ethnicity
Our study provides consistent evidence for racial/ethnic differences in humoral immune response following rubella vaccination.
However, in the conclusion section, the authors note that some caution may be required in interpreting such results from the angle of clinical relevance:
although it would be difficult to interpret our findings [...] in terms of possible protection from infection [... we] speculate that the higher neutralizing antibody levels observed for African-Americans (compared to Caucasians and/or Hispanics) in our study may potentially denote genetic and racial differences in the long-term immunity and protection following vaccination
A 2007 review (also on genetic variations) mentions two studies for observed ethnic differences in vaccine response; both studies found differences in the response to measles vaccine one on Bedouin vs Jewish Israelis response (to the MMR combo):
Seroconversion to measles was 99% in Bedouin and 79% in Jewish children (P < 0.01), and that to mumps and rubella was 92 to 100% in both groups [For the latter two presumably not a statistically significant difference; P not reported in the abstract.] Measles neutralizing antibody titers were higher in Bedouin (333 +/- 39 mIU/ml) than Jewish (122 +/- 60 mIU/ml) children (P < 0.002). [...] It is not known whether genetic differences or environmental exposure accounts for these differences.
the other on Inuit/Inuit vs Caucasian:
Native (Innu and Inuit) schoolchildren (n = 253) had a significantly higher seropositive rate (83%) after a single dose of measles vaccine compared to Caucasian (n = 353) children (76%; p = 0.025),
One interesting result to highlight perhaps is that a 2016 study on influenza vaccine (components) found that the ethnic/race differences were mainly found in the younger groups:
Race-related differences were caused by samples from younger African Americans, while results obtained with samples of aged African Americans were similar to those of aged Caucasians.