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Full Frame camera sensors will capture as much as 5x the amount of light for the same exposure time as an APS-C sized ("cropped") sensor on a DSLR.
But cameras with APS-C sized sensors can still do a good job of it. That also includes many recent mirrorless cameras that are not Full Frame.
This shot was taken with a Canon EOS-M (version 1) which costs around $199 these days or less. The lens used was a bright f/1.4L lens but a very cheap $99 EF-M 22mm f/2 STM lens can do much the same. This image shows a LESS bright portion of the Milky Way.
The Milky way is usually much more visible away from city lights although this shot was taken on the outskirts of a large city next to the ocean.
Compact cameras tend to have very small sensors on them and are either not sensitive enough to see the Milky Way or they produce too much noise.
Get a FREE program such as Stellarium and download it. Use it to determine when the BRIGHT areas of the Milky Way are likely to be visible.
Alternatively, by Star Walk for your iPod/iPhone/iPad and use that out in the field to determine where objects of interest are... including the brighter areas of the Milky Way.