Is an opamp that has a voltage range up to input voltage possible?
Yes there is - rail-to-rail input opamps. However, most of them tend to be rather low supply voltage ones - most of them around 5V, using CMOS technology rather than bipolar. One example would be the
LMV321/LMV358/LMV324, a low-voltage rail-to-rail input and output version of the
LM358/LM324 with the LMV321 being the single opamp version.
You will get better results if you increase the voltage across your current sense resistor so errors due to the opamp input offset voltage are reduced. If you don't mind getting more specialized parts you might want to take a look at some difference amplifiers - they're basically opamps which have the gain setting resistors built-in and closely matched.