We consider Lord Mahavishnu as having ten avatars because they descend to this material plane (Earth).
Doesn't that apply to Lord Ayyappa who is the son of Lord Shiva & Vishnu? Shouldn't he be an avatar of Lord Vishnu?
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The difference between a descended avatāra and a foolish embodied being (dehī) is that avatāra does not forget his spiritual identity in the process and maintains a strong connection with the spiritual plane. E.g. in the Kurukšētra battle Krišna was hit by all kinds of divine weapons, including one that Arjuna could not possibly deflect, but Krišna did not suffer. He could just absorb the weapons, himself being unarmed. Likewise, any avatāra is expected to manifest at least some exceptional supernatural prowess.
In Chaitanya-Charitāmrita 2.20.365 Sanātana Goswāmī asks a tongue-in-cheek question "Who is the avatāra in Kali age?". The question is amusing, because he is asking it to that avatāra himself and they both know that. However, the Master answers:
“O Sanātana, give up your intelligent tricks. Listen to the description of the šaktyāveša-avatāras.
To summarize his answer, an avatāra is recognized by the principal (mukhya) and superimposed (gauna) symptoms. In an avatāra, the manifestation of (divine) energy is direct (sākšāt-šaktye), if it is just a semblance (ābhāse), the person is called "vibhūti" instead. The symptoms to be looked for are described in the šāstras by muni Vjāsa (primary symptoms: name, skin color, dress, place of residence; divine energies: knowledge, love, the powers to create and maintain worlds and to protect and destroy villains; superimposed activities: killing of demons, creation, maintenance and destruction of the worlds and other). He also goes on to mention some avatāras and to describe the qualities and pastimes of Krišna in detail.
while we have not discovered a description in šāstras of Lord Ayyappa as an avatar of Lord Mahavishnu, we cannot accept him as such.
Ayyappa isn't an Avatar of Vishnu, rather his son in union with Shiva.
Quoting the official website of Shabarimala
Manikantan, an alias of Ayyappa was born of Hari (Lord Vishnu) and Hara (Lord Shiva), with Hari assuming the form of a female (Mohini). Hence Ayyappan is also named as Hariharasutan (Sutan meaning Son).
Where is the proof that Ayyappa is son of Hari- Hara or for that matter Hari or Hara or even whether he is an incarnation of siva or vishnu or dattatreya? These are mere folk lores which have no basis in any Vedas, ithihasa, 18 main puranas or upa puranas.
For your kind information, there is no mention or reference of Ayyappa by any of the Acharyas before 16th century even by any renowned saiva or sakta acharyas forget about vaishnava acharyas.
My intention is not to offend any one, but where is the proof from the valid hindu scriptures. There are none which indicate even remotely the Ayyappa story.
Ayyappa cult or story is more of a modern origin which has absolutely no valid proof in any Vedas, ithihasa, 18 main puranas or upa puranas.
Ayyappan has no reference in any scripture.No sexual intercourse happened b/w Lord Shiva and Lord Mohini. We can find it in Srimad Bhagavatam 8th Canto.
Since Yudhisthira is described as both an incarnation of Yama/Dharma and Indra, Yama/Ayyappan is an incarnation of Indra. Thus, Ayyappa is only an incarnation of Vishnu through Indra, which is pretty indirect.
Ayyappa/Yama could have more than one father like Kartikeya or Prachetasa.