Vedas clearly state about multiple rebirth in many places.
Here are some relevant quotes:
Kathopanishad of Krishna YajurVeda:
हन्त त इदं प्रवक्ष्यामि गुह्यं ब्रह्म सनातनम् ।
यथा च मरणं प्राप्य आत्मा भवति गौतम ॥ ६॥
योनिमन्ये प्रपद्यन्ते शरीरत्वाय देहिनः ।
स्थाणुमन्येऽनुसंयन्ति यथाकर्म यथाश्रुतम् ॥ ७॥
2-II-6. I will describe to thee, O Gautama, this secret ancient Brahman and also what becomes of the Self after death.
2-II-7. Some jivas enter the womb for assuming bodies; others go into the unmoving, in accordance with their karma and with their knowledge.
BrihadAranyaka Upanishad of Shukla YajurVeda:
ज॒यन्ति, ते॒ धूम॒मभिस॒म्भवन्ति,धूमा॒द्रा॒त्रिॅ रा॒त्रेरपक्षीयमाणपक्ष॒म् अपक्षीयमाणपक्षा॒द्या॒न्ष॒ण्मा॒सान्दक्षिणा॒दित्य॒ ए॒ति,
मा॒सेभ्यः पितृलोकं॒, पितृलोका॒च्चन्द्रं॒;
ते॒ चन्द्रं॒ प्रा॒प्या॒न्नं भवन्ति;ता॒ स् त॒त्र देवा॒ य॒था सो॒म रा॒जानम्
आ॒प्यायस्व,अ॒पक्षीयस्वे॒ति एव॒मेना स् त॒त्र भक्षयन्ति ।
ते॒षां यदा॒ त॒त्पर्यवै॒ति अ॒थेम॒मेवा॒काश॒मभिनि॒ष्पद्यन्त,
आकाशा॒द्वायुं॒,वायो॒र्वृष्टिं॒,वृष्टेः॒ पृथिवीं॒; ते॒ पृथिवीं॒ प्राप्या॒न्नं भवन्ति; ततो योषाग्नौ जायन्ते । त॒ एव॒मेवा॒नुप॒रिवर्तन्ते॒ ।अथ य॒ एतौ॒ प॒न्थानौ न॒ विदुः॒ ते॒ कीटाः॒, पत॒ङ्गा,य॒दिदं॒ दन्दशू॒कम् ।
VI-ii-16: While those who conquer the worlds through sacrifices, charity and austerity, reach the deity of smoke, from him the deity of the night, from him the deity of the fortnight in which the moon wanes, from him the deities of the six months in which the sun travels southward, from them the deity of the world of the manes, and from him the moon. Reaching the moon they become food. There the gods enjoy them as the priests drink the shining Soma juice (gradually, saying, as it were), ‘Flourish, dwindle’. And when their past work is exhausted, they reach (become like) this ether, from the ether air, from air rain, and from rain the earth. Reaching the earth they become food. Then they are again offered in the fire of man, thence in the fire of woman, whence they are born (and perform rites) with a view to going to other worlds. Thus do they rotate. While those others who do not know these two ways become insects and moths, and these frequently biting things (gnats and mosquitoes).
Chandogya Upanishad of SamaVeda:
तद्य इह रमणीयचरणा अभ्याशो ह यत्ते रमणीयां
योनिमापद्येरन्ब्राह्मणयोनिं वा क्षत्रिययोनिं वा वैश्ययोनिं
वाथ य इह कपूयचरणा अभ्याशो ह यत्ते कपूयां
योनिमापद्येरञ्श्वयोनिं वा सूकरयोनिं वा
चण्डालयोनिं वा ॥ ५.१०.७॥
V-x-7: Among them, those who have good residual results of action here (earned in this world and left as residue after the enjoyment in the region of the moon), quickly reach a good womb, the womb of a
Brahmana, or of a Kshatriya or of a Vaisya. But those who have bad residual results of action quickly reach an evil womb, the womb of a dog or of a hog or of a Chandala.
So, Vedas not only suggest but also clearly describe concepts of multiple Rebirths.