Axion is a promising dark matter candidate as well as a solution to the strong charge-parity (CP) problem in quantum chromodynamics (QCD). We describe a new concept for SmallSat Solar Axion and Activity X-ray Telescope (SSAXI) to search for solar axions or axion-like particles (ALPs) and to monitor solar activity over a wide dynamic range. SSAXI aims to unambiguously identify X-rays converted from axions in the solar magnetic field along the line of sight to the solar core, effectively imaging the solar core. SSAXI employs Miniature lightweight Wolter-I focusing X-ray optics (MiXO) and monolithic CMOS X-ray sensors in a compact package. The wide energy range (0.5 - 5 keV) of SSAXI can easily distinguish spectra of axion-converted X-rays from solar X-ray spectra, while encompassing the prime energy band (3 - 4.5 keV) of axion-converted X-rays. The high angular resolution (30 arcsec) and large field of view (40 arcmin) in SSAXI will easily resolve the enhanced X-ray flux over the 3 arcmin wide solar core while fully covering the X-ray activity over the entire solar disc. The fast readout in the inherently radiation tolerant CMOS X-ray sensors enables high resolution spectroscopy over a wide dynamic range with a broad range of operational temperatures. We present multiple mission implementation options for SSAXI under ESPA class. SSAXI will operate in a Sun-synchronous orbit for 1 yr preferably near a solar minimum to accumulate sufficient X-ray photon statistics.