If you shoot images for HDR you need to take bracketed shots at certain EV increments (mostly 1-2 EV steps are used). Many cameras support an AEB (Automatic Exposure Bracketing) mode.
Here is a list of features that can make your camera more or less easy to use for HDR photography.
Allowing EV step increments from 1/3 to at least 1 EV. Allowing up to 2 or 3 EV step increments is better. If the camera does not allow more than 2/3 step increments with a maximum of 3 shots, then the maximum EV range will be 1.4, which is insufficient for HDR photography.
Making it possible to take at least 5 auto-bracketed shots is better, but even some semi pro cameras only allow 3 shots. There are in principle two ways to break that barrier the manufacturers build into their firmware (we don't see any technical reasons at all, this is mostly to make a difference to the so-called pro cameras):
Allowing more than the old historic 30 seconds. If you use AEB in dark places, 30 seconds are often already too short (think of 9 exposures at 2EV steps). If the camera gets too noisy at longer exposure times the manufacturer should specify this and not prevent the user from using these longer shutter speeds.
There are two kinds of shake to consider:
Cameras with a mirror often add more shake because of the mirror slap. Try to shoot with a long lens (e.g. 200mm) from a tripod and you may already see major misalignments. At wider apertures, you may not easily see it but still get a lower quality image.
We shoot often AEB shots handheld. In such case, we use the camera or lens image stabilization to reduce shake. For AEB, we often use 1 to 1 1/3 EV.
Note: You should be able to use mirror lockup, live view and AEB all at the same time. Some cameras allow AEB and mirror lookup only as alternatives.
The faster you can shoot the less movement occurs in the scene. This is mainly important if you use AEB handheld. On the tripod it can be a mixed blessing because of shake introduced by the higher frame rate.
Not all cameras use their maximal frame rate also for AEB.
It would be ideal if the camera would only use AF for the first shot and then keep it constant. This could speed up the AEB operation and also keep the focus across all shots. Not many cameras do this, though.
Using manual focus avoids this problem altogether.