Ms Apruzzese said she wanted the health secretary to pay the charity a visit and witness the impact of its 12-week equine-assisted learning programmes "first-hand".
"We need to do more," she said. "Children and families wait years for the right support and in the meantime children are suffering."
But a spokesperson for the Department of Health said they cannot facilitate a visit at this time.
Ms Apruzzese said watching "ground-breaking" changes in children brought about by interaction with horses was "emotional".
In some instances, she said behaviours such as self-harm stopped all together.
She added that the NHS needs to look beyond "traditional therapies" and help organisations like Horses for Wellbeing to "reach more children before they hit crisis point".
Having lost both parents, nine-year-old Nancy and her sister now live with a foster family.
Nancy said being with horses "helps her a lot" and she had a "giant connection" with her favourite horse Marilyn.
"I really think she knows that I have a lot of things going on and I know that she has a lot of things going on too," she added.
Olivia arrived at Horses for Wellbeing highly anxious and selectively non-verbal but her favourite horse, Archie, helped her to start talking, her father, Asa, said.
"Since she's been coming here, she's gained more confidence, she's learning how to interact with people," he added.
"When she leaves here, she's got a big smile on her face - it's one of her happy places.
"The transformation, from our point of view, is amazing."
The charity said it helps schools and families with finances where possible through public donations, grants and sponsorship.
Belmont School, a special education school in Cheltenham, has three groups of children enrolled on the programme.
The school's pupils have complex needs and have often experienced trauma.
Its mental health lead, Kirstin Eccles, said the programme enables students to access the NHS' five steps to mental wellbeing "all in one morning".
She said it was "very special" to see the changes in children achieved at Horses for Wellbeing.
An NHS spokesperson said they are "seeing record demand for mental health support", with a 60% increase in young people accessing services compared to pre-pandemic.
"In response, the NHS has significantly boosted the children and young people's mental health workforce, as well as increased access to mental health support teams in schools so we can provide more support at an earlier stage," they added.