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I have a vector shapefile containing all of a state's counties. I also have a raster file with the statewide crop land use.

I need to calculate, for each individual county, the total acres of croplands.

I'm using ArcGIS 10.3

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The Zonal Statistics Tool is exactly what you need:

Calculates statistics on values of a raster within the zones of another dataset.

Depending on what kind of output you need, you may want Zonal Statistics As Table instead:

Summarizes the values of a raster within the zones of another dataset and reports the results to a table.

Both are part of the Spatial Analyst Extension.

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If you have Spatial Analyst extension, you need to use Tabulate Area tool. This is exactly what tabulate area tool can do. It calculates the area of each land use within the each zone boundary (county in your case).

The zone can be either raster or vector. The unit of the calculated area is decided based on the unit of the projection used. If the projection uses meter unit the output will be in square meter, you need to consider that during calculation.

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