No; it is not possible to hold such a belief without being considered either heretical, or at least withdrawn from full communion with the Catholic Church.
On 29 June 1998, the Congregation for the Doctrine of the Faith issued a document which discussed what teachings of the Catholic Church must be held by all Catholics. Section 11 of this document provides a list (not exhaustive) of such teachings.
The document was a commentary on a "Profession of Faith" which the Church requires of all who exercise an official Church function—including pastors of churches, teachers of theology, and others. The teachings are grouped by the CDF under three headings, depending on which paragraph of the Profession they refer to (see sections 5, 6, and 10):
[Doctrines of the first paragraph] are contained in the Word of God, written or handed down, and defined with a solemn judgment as divinely revealed truths either by the Roman Pontiff when he speaks 'ex cathedra,' or by the College of Bishops gathered in council, or infallibly proposed for belief by the ordinary and universal Magisterium. ... These doctrines require the assent of theological faith by all members of the faithful. Thus, whoever obstinately places them in doubt or denies them falls under the censure of heresy, as indicated by the respective canons of the Codes of Canon Law.
[Doctrines of the second paragraph include] all those teachings belonging to the dogmatic or moral area, which are necessary for faithfully keeping and expounding the deposit of faith, even if they have not been proposed by the Magisterium of the Church as formally revealed. ... Whoever denies these truths would be in a position of rejecting a truth of Catholic doctrine and would therefore no longer be in full communion with the Catholic Church.
[Doctrines of the third paragraph include] all those teachings on faith and morals - presented as true or at least as sure, even if they have not been defined with a solemn judgment or proposed as definitive by the ordinary and universal Magisterium. Such teachings are, however, an authentic expression of the ordinary Magisterium of the Roman Pontiff or of the College of Bishops and therefore require religious submission of will and intellect. ... A proposition contrary to these doctrines can be qualified as erroneous or, in the case of teachings of the prudential order, as rash or dangerous ...
According to the document,
To the truths of the first paragraph belong the articles of faith of the Creed, the various Christological dogmas and Marian dogmas
However, it does not specify precisely what those dogmas are. It refers, however, to the Denzinger-Schönmetzer Handbook of Creeds, Definitions, and Declarations on Matters of Faith and Morals, specifically to the sections regarding the papal declarations on the Immaculate Conception and the Assumption.
Thus, Catholics are certainly obliged to believe that Mary was free of original sin. I do not see a specific declaration that Mary was also free of personal sin; however, this belief appears to have been taken for granted. For example, the Apostolic Constitution Ineffabilis Deus, in which Pope Pius IX promulgated the belief in the Immaculate Conception, describes Mary as "absolutely free of all stain of sin"; and Pope Pius XII's encyclical letter Mystici Corporis Christi describes her (paragraph 110) as "free from all sin, original or personal".
Further, Thomas Aquinas discusses the question in the Summa Theologica (Third Part, Question 27, Article 4). He says that Mary
would not have been worthy to be the Mother of God, if she had ever sinned. First, because the honor of the parents reflects on the child, according to Prov. 17:6: "The glory of children are their fathers": and consequently, on the other hand, the Mother's shame would have reflected on her Son. Secondly, because of the singular affinity between her and Christ, who took flesh from her: and it is written ( 2 Cor. 6:15): "What concord hath Christ with Belial?" Thirdly, because of the singular manner in which the Son of God, who is the "Divine Wisdom" (1 Cor. 1:24) dwelt in her, not only in her soul but in her womb. And it is written (Wis. 1:4): "Wisdom will not enter into a malicious soul, nor dwell in a body subject to sins." We must therefore confess simply that the Blessed Virgin committed no actual sin, neither mortal nor venial.
If this is not a "doctrine of the first paragraph", then, it appears to be at least a "doctrine of the second paragraph", to be held by all Catholics.
I am unaware of any organized groups of self-proclaimed Catholics or former Catholics who have adopted such a belief corporately.