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Cryogenic environments benefit ion trapping experiments by offering lower motional heating rates, collision energies, and an ultrahigh 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 cryopackage assembly that enables easy handling while creating a UHV environment for the ions. The trap cryopackage 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.