Lord Shiva created Lord Brahma and then taught him the Vedas at the beginning of a kalpa (aeon), so that the process of creation can begin. This act establishes Lord Shiva as the Veda Karta (Creator of Vedas) and their ultimate origin.
To be more precise, Lord Shiva created all eighteen Vidyas (disciplines of knowledge), which include the four Vedas (Rig, Yajur, Sama, and Atharva).
There are several evidences:
~Śvetāśvatara Upaniṣad (6.18)
यो ब्रह्माणं विदधाति पूर्वं
यो वै वेदान्श्च प्रहिणोति तस्मै ।
तं ह देवमात्मबुद्धिप्रकाशं
मुमुक्षुर्वै शरणमहं प्रपद्ये ॥
Meaning:
He who created Brahmā in the beginning,
He who delivered the Vedas unto him —
That Deva (God), Śiva, who illuminates the intellect of the soul,
To Him I, seeking liberation, take refuge.
~Mahābhārata, Śānti Parva (349.65)
शिवादेव समुत्पन्ना वेदाः साङ्गा सनातनाः
Meaning:
From Śiva alone arose the eternal Vedas along with their auxiliaries.
~Liṅga Purāṇa (1.55.24)
शिववक्त्रविनिष्क्रान्ता वेदाः साङ्गाः सनातनाः
Meaning:
The eternal Vedas, along with their limbs, emerged from the mouth of Śiva.
~ Kūrma Purāṇa (1.12.129)
वेदान् सृजति देवेशः शिवो लोकमहेश्वरः
Meaning:
The Supreme Lord of the worlds, Śiva, created the Vedas.
Furthermore, It is believed that at the end of his cosmic dance (Tandava), Lord Shiva struck his drum (Damaru) 14 times. These 14 sounds formed the foundation of Sanskrit grammar and the sonic vibrations from which the Vedic hymns were realized.