One of the problems with interpolate() in Pandas is that it only works on continuous data. Using ffill(), you can fill objects. Using gropuby in iteration, we can fill in missing categorical data / object type data cells in our dataframe based on subsets.

interpolate_missing_object_values.py

import pandas as pd, numpy as np

data = [["blabla", "234234234", "yoyoyo", "Super Store235"],
        [np.nan, np.nan, np.nan, "Super Store"],
        [np.nan, np.nan, np.nan, "Super Store"],
        ["yo yo yo", 456, 789, "Super Store"],
        [np.nan, np.nan, np.nan, "Super Store"],
        [np.nan, np.nan, np.nan, "Super Store"],
        [123, 456, 789, "Super Store2"],
        [np.nan, np.nan, np.nan, "Super Store2"],
        [np.nan, np.nan, np.nan, "Super Store2"],
        [np.nan, np.nan, np.nan, "Super Store2"],
        [np.nan, np.nan, np.nan, "Super Store2"],
       ]

df = pd.DataFrame(data, columns=["county_number", "county_district", "random_num", "store"])

for group_label, group_df in df.groupby("store"):

    df[df["store"] == group_label] = df[df["store"] == group_label].sort("county_number").ffill()