If my husband has a custody order stating he gets to claim his daughter every year as a dependent, can he allow me to claim her as a dependent if we file separate?

Edited 2/25/18

A custodial parent always has the option of allowing the non-custodial parent to claim the dependency exemption and accompanying Child Tax Credit. 

He must furnish the other parent with a Form 8332: Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent.

In the case of someone who is not the biological parent, the Tiebreaker rules always benefit a parent.

Tiebreaker rules.   To determine which person can treat the child as a qualifying child to claim the six tax benefits just listed, the following tiebreaker rules apply.

• If only one of the persons is the child's parent, the child is treated as the qualifying child of the parent.
• If the parents file a joint return together and can claim the child as a qualifying child, the child is treated as the qualifying child of the parents.
• If the parents don't file a joint return together but both parents claim the child as a qualifying child, the IRS will treat the child as the qualifying child of the parent with whom the child lived for the longer period of time during the year. If the child lived with each parent for the same amount of time, the IRS will treat the child as the qualifying child of the parent who had the higher adjusted gross income (AGI) for the year.
• If no parent can claim the child as a qualifying child, the child is treated as the qualifying child of the person who had the highest AGI for the year.
• If a parent can claim the child as a qualifying child but no parent does so claim the child, the child is treated as the qualifying child of the person who had the highest AGI for the year, but only if that person's AGI is higher than the highest AGI of any of the child's parents who can claim the child. If the child's parents file a joint return with each other,

So he can release the dependency exemption to me and her mom can't do anything about it? Since he is the one that gets to claim her, he can allow me, the step-mom, to claim her?

Unless the child lived with you and your husband for the entire year, you may not claim the dependency exemption because you are not a blood relative. In addition, you would have had to provide more than half her support.  The original answer misinterpreted the facts of the situation, and assumed the child is your biological daughter.  It was a confusing question. @TurboTaxColeen - check your answer here.  The parent writing the question is not the mother of the child.

She doesn't live with us full time but the court order states that my husband gets the dependency exemption every year, so can't he release that exemption to me?

No, he can't.  You are not a blood relative, so one of the rules is that the dependent must have lived with you the entire year.  There are two types of dependents for tax purposes: qualifying children and qualifying relatives.  Your step-daughter is not a qualifying child because she is not a blood relative.  To claim a dependency exemption under qualifying relative rules, she must have lived with you for the entire year.

Form 8332, referenced by @TurboTaxColeen only provides for the release of the exemption to the noncustodial parent of the child - not a non-relative.

<a rel="nofollow" target="_blank" href="https://www.irs.gov/forms-pubs/form-8332-release-revocation-of-release-of-claim-to-exemption-for-child-by-custodial-parent">https://www.irs.gov/forms-pubs/form-8332-release-revocation-of-release-of-claim-to-exemption-for-child-by-custodial-parent</a>