"Our Vicar has told my husband that he will not be allowed to 'give' our daughter away in the well known and traditional way due to this not being in the Liturgy of the Anglican church these days. Is he correct?"
There are two issues – what is permissible and how decisions are made regarding the options.
There are two legal forms of service that can be used for a marriage and in both of them the “giving away” is possible. The modern services are called “Common Worship” and the traditional services are known as the 1662 Book of Common Prayer (BCP).
In Common Worship the “giving away” is optional. The decision should be made jointly but in the last resort the decision rests with the officiating minister but he ought to “give great weight to the parties’ wishes” (quoted from Anglican Marriage – A Guide to the Law for the Clergy paragraph 14.2)
In the Book of Common Prayer the “giving away” is strictly speaking obligatory, not optional.
The decision about whether to use the modern or traditional service should be made between the officiating minister and the couple, but if they cannot agree the decision rests with the Bishop (Canon B3(4)).
Common Worship (modern)
There is no mention of “giving away” in the text of the modern service but it is included in the notes which may be why your Vicar has missed it. Most churches use their own booklets or sheets but the official book is called “Pastoral Services” and the relevant note is on page 133 note 6.
Though it is called “giving away” the words refer to “bringing”. The relevant part of note 6 reads:
6 'Giving Away'
This traditional ceremony is optional. Immediately before the couple exchange vows the minister may ask:
Who brings this woman to be married to this man?
The bride's father (or mother, or another member of her family or a friend representing the family) gives the bride's right hand to the minister who puts it in the bridegroom's right hand....
1662 Book of Common Prayer (traditional)
Then shall the Minister say,
Who giveth this woman to be married to this man?
Then shall they give their troth to each other in this manner.
The Minister, receiving the Woman at her father's or friend's hands, shall cause the Man with his right hand to take the Woman by her right hand, and to say after him as followeth.
Troth is an old word meaning “truth”, “loyalty” or “fidelity” and the man ends the following declaration by saying “thereto I plight thee my troth” - meaning he pledges his loyalty and fidelity to his wife. The wife ends her response by saying “and thereto I give thee my troth”. That is, in response to his pledge of faithfulness she gives her faithfulness to him.
Text of the 1662 Marriage service