Are there any publicly available telescope that offers the sensitivity to observe the stars surrounding Sgr A*? If not, is there any raw data available that I could download?
Observations of stars near Sgr A* are done in the near-infrared, usually with adaptive optics. Most of these are done by two groups: Andrea Ghez's group at UCLA, using the Keck Telescopes in Hawaii, and Reinhard Genzel's group at MPE (Max-Planck-Institute for Extraterrestrial Physics), using the European Southern Observatory's Very Large Telescope (VLT) in Chile.
Both telescopes have public data archives, so in principle you can retrieve the raw data for these observations, as long as the observations are at least a year old (which most of them are). You need to know which are the relevant instruments; NACO has been used for most of the images at the VLT, and I think NIRC2 has been used at Keck. (There are also spectroscopic observations using both telescopes.)
So you could, for example, search using the NACO-specific search form in the ESO archive. I would suggest entering "Sgr A*" in the "Target name" field, restricting the "Search Box" field to something like 30 arc seconds (change the default "00 10 00" to "00 00 30"). Note that these observations involve lots of short exposures in different filters, and with no other restrictions on the search, you will get more than 10,000 hits.
(In order to actually retrieve the data, you will need to register as a user of the archive, which is free.)
Something similar might work for the Keck Observatory Archive, though I've never really used that.
You'll need to be familiar with the reduction and analysis of near-IR imaging to get anything from the data, of course. ESO does provide "pipeline" software that can do some of the reduction (search on their site).
You can, in principle, apply to use the VLT yourself, but unless you really know what you're doing, it's very unlikely you'd get time in competition with, say, Genzel's group. Also, NACO has been taken off the VLT; its replacement instrument, ERIS, won't be available until sometime in 2020. (You can't generally apply for Keck time unless you're at one of its member institutions.)
Not much will be visible in optical wavelength as the region will be too bright or nothing will be visible. You can observe in X-ray practically as black holes emit synchrotron radiation and that is visible in x-ray region pretty good. So you can write a proposal to CXO and make them understand your need to observe them Sag A* and they will allot you time. Then you can process that.