Cryogenic environments benefit ion trapping experiments by offering lower motional heating rates, collision energies, and an ultra-high vacuum (UHV) environment for maintaining long ion chains for extended periods of time. Mechanical vibrations caused by compressors in closed-cycle cryostats can introduce relative motion between the ion and the wavefronts of lasers used to manipulate the ions. Here, we present a novel ion trapping system where a commercial low-vibration closed-cycle cryostat is used in a custom monolithic enclosure. We measure mechanical vibrations of the sample stage using an optical interferometer, and observe a root-mean-square relative displacement of 2.4 nm and a peak-to-peak displacement of 17 nm between free-space beams and the trapping location. We packaged a surface ion trap in a cryo-package assembly that enables easy handling, while creating a UHV environment for the ions. The trap cryo-package contains activated carbon getter material for enhanced sorption pumping near the trapping location, and source material for ablation loading. Using $^{171}$Yb$^{+}$ as our ion we estimate the operating pressure of the trap as a function of package temperature using phase transitions of zig-zag ion chains as a probe. We measured the radial mode heating rate of a single ion to be 13 quanta/s on average. The Ramsey coherence measurements yield 330 ms coherence time for counter-propagating Raman carrier transitions using a 355 nm mode-locked pulse laser, demonstrating the high optical stability.