Background: The production of Be7 and Li7 nuclei plays an important role in primordial nucleosynthesis, nuclear astrophysics, and fusion energy generation. The He3(α,γ)Be7 and H3(α,γ)Li7 radiative-capture processes are important to determine the Li7 abundance in the early universe and to predict the correct fraction of pp-chain branches resulting in Be7 versus B8 neutrinos. The Li6(p,γ)Be7 has been investigated recently hinting at a possible cross section enhacement near the thershold. The Li6(n,H3)He4 process can be utilized for tritium breeding in machines dedicated to fusion energy generation through the deuteron-tritium reaction, and is a neutron cross section standard used in the measurement and evaluation of fission cross sections. Purpose: In this work we study the properties of Be7 and Li7 within the no-core shell model with continuum (NCSMC) method, using chiral nucleon-nucleon interactions as the only input, and analyze all the binary mass partitions involved in the formation of these systems. Methods: The NCSMC is an ab initio method applicable to light nuclei that provides a unified description of bound and scattering states and thus is well suited to investigate systems with many resonances and pronounced clustering like Be7 and Li7. Results: Our calculations reproduce all the experimentally known states of the two systems and provide predictions for several new resonances of both parities. Some of these new possible resonances are built on the ground states of Li6 and He6, and thus represent a robust prediction. We do not find any resonance in the p+Li6 mass partition near the threshold. On the other hand, in the p+He6 mass partition of Li7 we observe an S-wave resonance near the threshold producing a very pronounced peak in the calculated S factor of the He6(p,γ)Li7 radiative-capture reaction. Conclusions: While we do not find a resonance near the thershold in the p+Li6 channel, in the case of He6+p reaction a resonant S-wave state is predicted at a very low energy above the reaction threshold, which could be relevant for astrophysics and its implications should be investigated. We note though that this state lies above the three-body breakup threshold not included in our method and may be influenced by three-body continuum correlations.