Short answer. Not under this provision as written today. Rule 93, Section 1 of the Rules of Court no longer governs guardianship of minors at all; A.M. No. 03-02-05-SC, the 2003 Rule on Guardianship of Minors, now governs that specifically, so who may petition on a minor's behalf is answered by that later rule, not by this one.
What the law says
or the minor himself if fourteen years of age or over, may petition the court having jurisdiction for the appointment of a general guardian for the person or estate, or both, of such minor or incompetent
Rule 93, Section 1 — Who may petition for appointment of guardian for resident. Read the full provision →
What this provision said as originally written
As enacted, Rule 93, Section 1 let a broad set of people petition for a guardian on behalf of a resident minor who had no parent or lawful guardian, and it also gave the minor a route to act personally: or the minor himself if fourteen years of age or over, may petition the court having jurisdiction for the appointment of a general guardian for the person or estate, or both, of such minor or incompetent. Under that original text, a fourteen-year-old or older, which covers your fifteen-year-old, could file the petition without needing someone else to do it on his behalf.
Why that original text is no longer the rule for minors
This provision has since been overtaken specifically as to minors. A.M. No. 03-02-05-SC, the Rule on Guardianship of Minors, effective May 1, 2003, amended this and related rules of the Rules of Court precisely on that subject, and it carves guardianship of minors out of the older framework. The older Rules of Court provisions, including this one, continue to govern guardianship of incompetents who are not minors, but for a minor ward, the 2003 Rule is now the operative one, not the text quoted above.
What this means for a fifteen-year-old orphan today
Because your situation concerns a minor, not an incompetent adult, the question of who may petition for your guardian, and on what age threshold, is governed by the 2003 Rule on Guardianship of Minors rather than by this older provision. This page's verified text does not extend to that later rule's specific terms, so it cannot responsibly tell you what age threshold or petitioning process the 2003 Rule sets for a minor acting on his own behalf. What it can tell you clearly is that relying on the fourteen-years-or-over language quoted above, as though it still controlled guardianship of minors, would be relying on a rule that no longer governs that situation.
Why this distinction matters in practice
The practical risk of treating the original text as still controlling is that it could lead a minor's family, or the minor himself, to petition under the wrong framework, or to assume the fourteen-year threshold still applies when the 2003 Rule may set its own conditions. Since guardianship of a minor ward is now governed by that later rule rather than this one, anyone in your situation is better served by working from the 2003 Rule on Guardianship of Minors directly, rather than treating this older provision as the current source of the answer.