Is it permissible to track a husband and investigate him due to suspicion of infidelity, because things belonging only to the wife have disappeared from the house, or should the matter be left to God?

It is not permissible for you to spy on your husband, accuse him of theft, or of bringing women into the house without clear proof. The fundamental principle regarding a Muslim is their integrity. You must repel these suspicions and seek refuge with Allah from Satan, while maintaining a good opinion of a Muslim.