We present results from a new ultra-deep approximate to 400 ks Chandra observation of the SSA22 protocluster at z = 3.09. We have studied the X-ray properties of 234 z similar to 3 Lyman Break Galaxies (LBGs; protocluster and field) and 158 z = 3.09 Ly alpha Emitters (LAEs) in SSA22 to measure the influence of the high-density protocluster environment on the accretion activity of supermassive black holes (SMBHs) in these UV-selected star-forming populations. We detect individually X-ray emission from active galactic nuclei (AGNs) in six LBGs and five LAEs; due to small overlap between the LBG and LAE source population, ten of these sources are unique. At least six and potentially eight of these sources are members of the protocluster. These sources have rest-frame 8-32 keV luminosities in the range of L(8-32) (keV) = (3-50) x 10(43) ergs s(-1) and an average observed-frame 2-8 keV to 0.5-2 keV band ratio ( BR) of approximate to 0.8 (mean effective photon index of Gamma(eff) approximate to 1.1), suggesting significant absorption columns of N(H) greater than or similar to 10(22)-10(24) cm(-2). We find that the fraction of LBGs and LAEs in the z = 3.09 protocluster harboring an AGN with L(8-32 keV) greater than or similar to 3 x 10(43) ergs s(-1) is 9.5(-6.1)(+12.7)% and 5.1(-3.3)(+6.8)%, respectively. These AGN fractions are somewhat larger (by a mean factor of 6.1(-3.6)(+10.3); significant at the approximate to 95% confidence level) than z similar to 3 sources found in lower-density "field" environments. Theoretical models imply that these results may be due to the presence of more actively growing and/or massive SMBHs in LBGs and LAEs within the protocluster compared to the field. Such a result is expected in a scenario where enhanced merger activity in the protocluster drives accelerated galaxy and SMBH growth at z greater than or similar to 2-3. Using Spitzer IRAC imaging we found that the fraction of IRAC-detected LBGs is significantly larger in the protocluster than in the field (by a factor of 3.0(-1.3)(+2.0)). From these data, we constrained the median rest-frame H-band luminosity in the protocluster to be greater than or similar to 1.2-1.8 times larger than that for the field. When combined with our X-ray data, this suggests that both galaxies and SMBHs grew more rapidly in protocluster environments.