Zakat Must Be Paid by Yourself

Zakat is an obligation upon each individual who has wealth on which Zakat is compulsory. Zakat may be paid by such an individual directly or via a proxy/representative.

Therefore you may pay your Zakat yourself or instruct someone to pay it on your behalf. Although, your mother-in-law is not obligated to pay your Zakat, if she pays it gratuitously on your behalf with your prior consent then your Zakat is valid and discharged. The same applies to your husband's Zakat. Each of you should still calculate your individual Zakat annually, even though this is discharged by someone else on your behalf with your consent.

And Allah Ta'ala Knows Best.