Are self-composed supplications (du’as) allowed, or is that innovation?

Self-composed supplications are allowed, as long as they don't compete against a specific du'a the Sunnah prescribed for that moment. The Prophet ﷺ himself encouraged supplications that were not part of the Sunnah.

The Prophet ﷺ not only taught the Companions supplications but also encouraged them to make their own, and accorded them the highest acceptance. As authenticated by the hadith, Burayda may Allah be pleased with him said, “I entered the Masjid with the Messenger ﷺ, where a man was supplicating; ‘O Allah, I ask you by the fact that I testify, You are Allah, there is no god but You, the One, the Ultimate, who did not beget and was not begotten, and to whom no one is equal,’ and the Prophet ﷺ said, ‘By Him in whose hands is my soul, he has asked Allah by His Greatest name, which if He is asked by it He gives, and if supplicated, He answers.’” [Sahih Ibn Hibban 892]

The Companions, their students, and the scholars after them composed their own litanies of remembrance.

The one limit is this: a personal supplication becomes an innovation only when it competes against a specific du'a the Sunnah legislated for a particular time and place. Outside of that, supplicating to Allah in one's own way and one's own time is not an innovation.