Which group of currently active Hindus (not necessarily a sect) reject the least from the totality of Hindu scripture?
Yes, orthodox Hindus (Sri Vaishnavas) reject the least from Hindu scripture.
Neo-Vedantins/neo-Hindus (Vivekananda brand) reject the most from Hindu scripture. They reject everything except the Gita, Upanishads, and Brahma Sutras. This includes rejecting the dharma shastras (manusmriti), etc.
Vivekananda has said:
The Smritis and the Puranas are productions of men of limited intelligence and are full of fallacies, errors, the feelings of class and malice. Only parts of them breathing broadness of spirit and love are acceptable, the rest are to be rejected. The Upanishads and the Gita are the true scriptures; Rama, Krishna, Buddha, Chaitanya, Nanak, Kabir, and so on are the true Avatâras, for they had their hearts broad as the sky — and above all, Ramakrishna.
Arya Samaj also reject a lot. They reject many of the upanishads as man made. They reject many of the caste-by-birth verses in the Manusmriti as interpolations and/or interpret it in a different way. They also think the Rig Veda samhita is true hindu scripture and everything else is man made.
The Arya Samaj members reject other Hindu religious texts because they are not "revealed" works.
Arya samaj is also feminist and rejects "superstitions":
Among Dayananda's contributions were his promoting of the equal rights for women, such as the right to education and reading of Indian scriptures
Hindu priests discouraged the laity from reading Vedic scriptures, and encouraged rituals, such as bathing in the Ganges River and feeding of priests on anniversaries, which Dayananda pronounced as superstitions or self-serving practices.
The two sects before are considered reformist sects of Hinduism. Now I'll talk about the orthodox sects.
Orthodox Advaitins, followers of shankaracharya, accept everything except the Pancharatra agamas, which shankaracharya rejects in his brahma sutra commentary.
Madhvacharya accepts the Pancharatra agamas, but his interpretations of scripture deviate a lot from all other orthodox Vedantic sects.
And finally, Sri Vaishnavas accept almost everything from Hindu scripture, because Ramanujacharya cites this verse from the Mahabharata:
Itihasa puranabhyam vedam samupabrhmayet - Mahabharata (Adi Parva)
"One should interpret the Vedas with the help of the Itihasas and Puranas."
So they take smriti very seriously.
Jiva Goswami of the Gaudiya Vaishnava branch also cites this verse, and they also cite some Sri Vaishnava works, so Gaudiya Vaishnavism is also an orthodox sect.
All Vaishnava sects accept the Pancharatra Agamas.
Sri Vaishnavism accepts the whole Veda including Upanishads, all dharma shastras, all 18 puranas as well as the upapuranas, Ramayana, Mahabharata, Pancharatra agamas, and darshana sutras, among other texts.