Upvote:2
Maybe you are overthinking this?
I'm not aware of a biblical basis for the congregation to take a vow, specifically. However, we certainly have a responsibility as Christians to support each other and to educate each other in the faith. In particular, note the very basis of Baptism:
19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 teaching them to observe all that I have commanded you.
(Matthew 28:19-20a)
The expectation "to guide and nurture [name(s)] by word and deed, with love and prayer, encouraging them to know and follow Christ and to be faithful members of his church" is in essence the same as Christ's own command to "teach them to observe all that I have commanded you".
Thus, while the Bible may not specifically command us to take a vow to do this specific thing which Christ commanded, it is hardly unnatural for us to take such a vow.
If there is a question here, I think it should rather be whether Christians should take vows to obey God/Christ at all when not specifically commanded to do so. Consider marriage vows, confirmation vows, vows of office for pastors and church workers, etc..