Is it permissible to seek divorce from a husband who is excessively doubtful and untruthful, and who makes false accusations, given the presence of four children? And would the wife be sinful for doing so?

If the wife's situation with her husband is as you described, involving psychological suffering and feelings of humiliation due to his accusation of infidelity, then there is no blame upon her, from a legal perspective, for seeking divorce. However, one should not rush into it. Instead, he should be advised with wisdom and good admonition, reminded of Allah the Almighty, and warned of His punishment, while seeking help from the religious texts that caution against accusing innocent people. The intent is to be deliberate and work towards reconciliation, because the presence of children might make them victims in the event of divorce. Therefore, divorce should not be resorted to unless cohabitation becomes impossible and its benefit outweighs its harm. One should also avoid anything that raises suspicion, especially if the husband is paranoid, and avoiding situations of doubt is a religiously mandated requirement.