A person who dandles; someone who bounces or plays gently with a baby or small child.
From dandle + -er (the suffix that creates agent nouns—people who do something). This is a straightforward English word formation: verb + -er = 'one who verbs.'
Dandler is mostly theoretical—grammarians can build it perfectly (it follows English rules), but nobody really says 'the dandler dandled the baby.' We'd more likely say 'parent' or 'caregiver.' It shows how English can make words that are grammatically perfect but never actually get used.
Complete word intelligence in one call. Free tier — 50 lookups/day.