I was recently looking for the answer to this question and found something that was useful for what I was doing and thought it would be helpful to share. I ended up using a permutation importance module from the eli5 package. It most easily works with a scikit-learn model. Luckily, Keras provides a wrapper for sequential models. As shown in the code below, using it is very straightforward.
from keras.wrappers.scikit_learn import KerasClassifier, KerasRegressor
import eli5
from eli5.sklearn import PermutationImportance
def base_model():
model = Sequential()
...
return model
X = ...
y = ...
my_model = KerasRegressor(build_fn=base_model, **sk_params)
my_model.fit(X,y)
perm = PermutationImportance(my_model, random_state=1).fit(X,y)
eli5.show_weights(perm, feature_names = X.columns.tolist())