We report a rectangular charge density wave (CDW) phase in strained 1T-VSe$_2$ thin films synthesized by molecular beam epitaxy on c-sapphire substrates. The observed CDW structure exhibits an unconventional rectangular 4a{times}{sqrt{3a}} periodicity, as opposed to the previously reported hexagonal $4atimes4a$ structure in bulk crystals and exfoliated thin layered samples. Tunneling spectroscopy shows a strong modulation of the local density of states of the same $4atimessqrt{3}a$ CDW periodicity and an energy gap of $2Delta_{CDW}=(9.1pm0.1)$ meV. The CDW energy gap evolves into a full gap at temperatures below 500 mK, indicating a transition to an insulating phase at ultra-low temperatures. First-principles calculations confirm the stability of both $4atimes4a$ and $4atimessqrt{3}a$ structures arising from soft modes in the phonon dispersion. The unconventional structure becomes preferred in the presence of strain, in agreement with experimental findings.
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