Is it permissible for the bride to perform tayammum instead of wudu for the Sunnah prayer with her husband on their wedding night?

It is not permissible to perform tayammum (dry ablution) when water is available and one is able to use it, whether for a Sunnah prayer or a fard (obligatory) prayer. This is based on the Almighty’s saying: (and find no water, then perform tayammum with clean earth). Maintaining one’s appearance for the husband, even on the wedding night, is not an excuse to avoid using water.