How to calculate the current / power consumption of instrumentation amp with op amp as an impedance converter at REF input
To determine the power consumption of your LP358 based voltage buffer (I would not call that an impedance converter) you need to know two things:

the current consumption of the opamp itself, when there is no current flowing into or out of the output. You can find this number in table 6.5: \$I_{CC}\$ Supply current which is typically 54 uA. This is for a supply voltage of \$V_{CC}\$ = 5 V.

the current that flows out of the opamp's output. If there is a current flowing out of the output, then the opamp is supplying current to another circuit and that current needs to come from somewhere, it comes from the supply so this current adds to the current consumption of the opamp itself (the 54 uA).


In your circuit, the output current of the LP358 flows into the reference voltage input of the AD8227 so we'll have to look in the datasheet of the AD8227 what is connected there and how much current it takes. Table 2 shows what we need: REFERENCE INPUT, \$I_{IN}\$ has a typical value of 12 uA.
Now simply add both: 54 uA + 12 uA = 66 uA. Note that these are typical values so the actual value can be somewhat lower or higher. Depending on the circumstances the LP358 can consume up to 125 uA (max value in the datasheet).

Does the gain resistor of the AD8227 have an
impact on the current consumption?

The AD8227's datasheet only mentions figures for a Gain = 5. My guess is that the AD8227's current consumption might be affected by the Gain you choose. I highly doubt that the current into the reference voltage input depends on the Gain though. To be sure you need to measure it.