Of course, you can. A slogan or a tagline adds depth of meaning to a simple business logo. Once you have chosen the daycare logo design you like, click on it to go to the customization page and in the 'add text' field, add your business slogan.
Following the same procedure, you can add a slogan to your babysitter logo, preschool, childcare, or nanny services logo, basically any and all logo templates.
When adding the slogan to your daycare logo template, you also have the option to change the font size, style, color, and placement.