In short: No, we cannot do that at the moment.

We have photometric all-sky surveys that can map the entire sky (visible from the telescope location) during a night up to a certain brightness. But those only take images of the sky, not spectra (Zwicky Transient Facility and the planned Vera C. Rubin Observatory).

What we also have are integral-field spectrographs which can take 2D images with a twist: there is one image for every ~0.1nm from 480nm to 950nm. You take one exposure with the instrument and you get a stack of thousands of images. If you go through the stack at a fixed spatial position you get the spectrum. The problem is that the integral-field spectrograph with the largest field-of-view is already huge (it is called MUSE at is located at the Very Large Telescope). And its field-of-view is "only" 1 arcmin^2 (1 deg = 60 arcmin), which is by far too small for large surveys. If you wanted to image the whole sky each night with MUSE clones, you would need several millions of them. A single MUSE exposure is about ~5 GB in the end but there are intermediate data products which are about 10 GB, if I remember correctly.