Nikon lenses tend to have good close focus and macro modes.

Take a look at their new series of cameras:

4600, 5600, 5900, 7600 and 7900.

The price ranges from $174 to $356 for those cameras.

All will close-focus to 30cm (one foot).
All will macro focus to 4cm (1.5 inches).
..(that is the distance from the front of the lens to the target)

This should easily cover the range for photographing flowers.

With any camera, when focusing at those close ranges, it is advisable to use a tripod.
Photographing in well lit areas is also advisable, to increase the depth of field.