DS has had ongoing issues with both gross and fine motor and he has hypotonia and hypermobile joints (they go together). Look up hypotonia in Wikipedia.
He also has Developmental Coordination Disorder.
He does not have any obvious issues like sitting up in school. You can really see it when he is asked to do things like hop or jump. If she is 4, she should be able to hop up and down, jump forward easily with both feet together, jump off something like a couch without falling on her face or her legs collapsing on impact, hang from a bar with her hands, etc.
If she has hypotonia, it would also be hard for her to do things like hold a pencil properly. If she writes, she may not press down on the paper very hard.
In your shoes, I would ask the pediatrician for a referral for a physical and occupational therapy assessment (they will probably be done by 2 people in two different days). She may qualify for private therapy, or you can use the results to get the school district to do an evaluation. Depending on how low she scores, she could qualify for an IEP with the school district and get therapy free.
