Surely, Bhagwan Āditya is God of weather because he is called ऋतुकर्ता and it's because of him only we experience seasonal variation.
While, there are specific Gods and Goddesses for each weather and they reside in different Gardens(Vāṭikā-s) of Maṇidweepa(abode of Ambā),
So, Śrī Lalitā Mahā Tripurasundari is the Goddess of Weather
From Brahmāṇḍa Purāṇa :-
वसन्तर्तुर्महातेजा ललिताप्रियकिङ्करः ॥
पुष्पसिंहासनासीनः पुष्पमाध्वीमदारुणाः ।
पुष्पायुधः पुष्पभूषः पुष्पच्छत्रेण शोभितः ॥मधुश्रीर्माधवश्रीश्च द्वे देव्यौ तस्य दीव्यतः ।
Vasanta Ṛtu—the spring season, resides in Kalpakavāṭī of Maṇidweepa is the favourite servant of Lalitā. He is seated on a flowery throne. He is reddened due to intoxication by the liquor extracted from the honey of flowers. Flowers are his weapons and ornaments. He appears splendid with a flowery umbrella, Madhuśrī and Mādhavaśrī are the wives of that shining lord (Vasanta)
सन्तानवाटिकापालो ग्रीष्मर्तुस्तीक्ष्णलोचनः ।
ललिताकिङ्करो नित्यं तस्यास्त्वाज्ञाप्रवर्तकः ॥शुक्र श्रीश्च शुचिश्रीश्च तस्य भार्ये उभे स्मृते ।
Grīṣma Ṛtu:- The Summer Season is the defender of Santānavāṭikā having fiery eyes. He is the servant of Lalitā, continuously carrying out her commands. Śukraśrī and Śuciśrī remembered as his two wives.
स वर्षर्न्तुर्महातेजा विद्युत्पङ्गललोचनः ।
वज्राट्टहासमुखरो मत्तजीमूतवाहनः ॥जीमूतकवचच्छन्नो मणिकार्मुकधारकः ।
ललितापूजनध्यानजपस्तोत्रपरायणः ॥वर्तते विन्ध्यमथन त्रैलोक्याह्लाददायकः ।
नभःश्रीश्च नभस्यश्रीः स्वरस्वारस्वमालिनी ॥अम्बा दुला निरलिश्चाभ्रयन्ती मेघयन्त्रिका ।
वर्षयन्ती चिबुणिका वारिधारा च शक्तयः ॥वर्षन्त्यो द्वादश प्रोक्ता मदारुणविलोचनाः ।
Varṣartu:- the rainy season, occupies Haricaṇḍanavātikā of Maṇidweepa having great splendour. Lightning-streaks are his tawny eyes. He is highly resonant with thunderbolt as his boisterous laughter. He has haughty clouds for his vehicle. He is covered with clouds like coats of mail. He holds a jewel-set bow. He is absorbed in worshipping, eulogising, meditating upon and repeating the names of Lalitā. He is, O conqueror of Vindhya, one who gives great delight to all the three worlds.
His Śaktis are said to be twelve, viz. Nabhaśrī, Nabhasyaśrī, Svarasā, Svara, Mālinī, Ambā, Dulā, Nirali, Abhrayanī, Meghayantrikā, Varṣayantī, Cibuṇikā and Vāridhārā. They shower rain. They have eyes reddened due to inebriation.
मन्दारवाटिकायां तु सदा शरदृतुर्वसन् ।
तां कक्षां रक्षति श्रीमांल्लोकचित्तप्रसादनः ॥इषश्रीश्च तथोर्जश्रीस्तस्यर्तोः प्राणनायिके ।
ताभ्यां संजह्रतुस्तोयं निजोत्थैः पुष्पमण्डलैः ।
अभ्यर्चयति साम्राज्ञीं श्रीकामेश्वरयोषितम् ॥
Śaradṛtu:- the Autumn, stays permanently in the grove of Mandāra trees and protects that chamber. He is splendidly glorious. He delights the mind of the world. Iṣaśrī and Ūrjaśrī are the beloved ones of that season. They gather water and with clusters of flowers raised by himself, he worships dthe empress, the consort of Śrīkāmeśvara.
हेमन्तर्तुर्महातेजा हिमशीतलविग्रहः ।
सदा प्रसन्नवदनो ललिताप्रियकिङ्करः ॥निजोत्थैः पुष्पसंभारैरर्चयन्परमेश्वरीम् ।
पारिजातस्य वाटीं तु रक्षति ज्वलनार्दनः ॥सहःश्रीश्च सहस्यश्रीस्तस्य द्वे योषिते शुभे ।
Hemantartu:- Season of early winter, of great splendour has his physical body as cool as snow. His face is always delightful. He is the favourite servant of Lalitā. By means of clusters of flowers raised by himself, he worships the supreme goddess. He defends the park of Pārijāta. He subdues even fire. Sahaḥśrī and Sahasyaśrī are his two excellent consorts.
कदम्बवनवाट्यास्तु रक्षकः शिशिराकृतिः ॥
शिशिरर्तुर्मुनिश्रेष्ठ वर्तते कुम्भसम्भव ।
सा कक्ष्या तेन सर्वत्र शीशिरीकृतभूतला ॥तद्वासिनी ततः श्यामा देवता शिशिराकृतिः ।
तपःश्रीश्च तपस्यश्रीस्तस्य द्वे योषिदुत्तमे ।
ताभ्यां सहार्चयत्यंबां ललितां विश्वपावनीम् ॥
Śiśirartu:- the season of winter, guard of the grove of Kadamba Trees is of cool features, O excellent sage born of pot. Therefore, the entire surface of the land of that chamber is rendered cool by him. Therefore, Śrī Rāja Śyāmalā, the deity residing there is also of cool nature and features. His two excellent consorts are Tapaḥśrī and Tapasyaśrī. Along with them he worships Lalitā, the mother who sanctifies the universe.
Same way, the praṃāṇas can also be seen in other texts like Tripurā Rahasya, Lalitā Stavaratna of Bhagwān Dūrvāsā and Devī Bhāgwat
I hope this answered the question🙏🏻
Namastripurasundari🌺