Replacing an instrumentation amp with an op amp
The key word here to search for is "Instrumentation Amplifier". These have several important characteristics for use amplifying a load cell: Low offset voltage, low bias current, and high common-mode voltage range. They also have gain settable via a single external resistor.

You could theoretically find a regular opamp with those properties and build a differential input amplifier circuit, but you'd have to buffer the input signals to get low input impedance, and any mismatch in your components would contribute to error. In practice, it's much easier to use a premade instrumentation amp. INA125 is one such, but there are many alternatives.