Yes, 'I will ask Elliott to tell her ' is correct. The construction 'I'll have Elliott tell her' is also correct, but it has a slightly different connotation. It means that the speaker will arrange in some way for Elliott to tell her. It is a more indirect statement, and is likely to be used in a more formal or business context where there are more complex systems of protocol. It sounds like something that a boss might say. The form [have] + [person] + [infinitive of main verb] is a type of causative construction. Another example would be 'I'll have my secretary call you', meaning that I'll ask/arrange for my secretary to call you. This is an advanced level grammatical form which you need not worry about unless you need to take an English exam at C1 or above.