We report on the X-ray activity from Lyman Break Galaxies (LBGs) and Lyman Alpha Emitters (LAEs) in the z = 3.09 proto-cluster SSA22. Using a new ~400 ks Chandra exposure over the central region of the SSA22 proto-cluster, we detect individually in X-rays four LBGs and six LAEs. These sources have rest-frame 8--32 keV luminosities in the range of 4 X 10^43 to 10^45 ergs s^-1 and typical observed 2--8 keV to 0.5--2 keV band-ratios of 0.7--1.8, indicating that these sources range from highly-absorbed to Compton-thick AGNs. We compare the X-ray properties of the SSA22 proto-cluster LBGs and LAEs with equivalent lower-density samples found in the Chandra Deep Fields (CDFs). We find that the fractions of LBGs and LAEs hosting luminous AGNs, with 8--32 keV luminosities larger than 10^44 ergs s^-1, are suggestively elevated in the SSA22 proto-cluster compared with the field population in the CDFs. We constrain the enhanced level of accretion activity per galaxy to be < 7.6 times that observed in the field population. Using X-ray stacking analyses, we studied the remaining X-ray--undetected LBG and LAE source populations in the SSA22 field and find that the mean X-ray power from these systems are comparable to those found in the CDFs. This result suggests that star-formation and low-luminosity AGN activity is not significantly enhaced in the SSA22 proto-cluster environment.