Well, the Upanishads do not have the fully-developed seven chakras and kundalini system that is found in later medieval texts.
What they have is some hints and rudiments of it (however, this is not central to their teaching). They talk about special "nerves" (nāḍī नाडी) that carry energy. But of course, they might also mean blood vessels. As you know in Ayurveda, the first diagnosis is nāḍīparīkṣā (नाडीपरीक्षा), which is essentially checking the pulse, which as we know is found in blood vessels and not nerves. But in the Upanishads there does not seem to be an understanding of the strict differences between blood vessels and nerves. The Upanishads describe these nāḍīs not as objective fact, but as subjective experience. In other words, these nāḍīs cannot be found from dissection of a body, but they can be experienced as energy channels within the body by a living person. With our modern understanding of physiology, I would say that the nāḍīs are the subtle energy channels resulting from the interactive function of blood vessels and nerves, because these two do not function in isolation. They are part of a closed-loop control system.
The earliest text that talks about this is Chandogya Upanishad (8.6):
अथ या एता हृदयस्य नाड्यस्ताः पिङ्गलस्याणिम्नस्तिष्ठन्ति शुक्लस्य नीलस्य पीतस्य लोहितस्य
Now, there are these tiny "nerves" of the heart, which are tawny, white, blue/black, yellow and red.
... आदित्यस्य रश्मयः ... अमुष्मादादित्यात्प्रतायन्ते ता आसु नाडीषु सृप्ता आभ्यो नाडीभ्यः प्रतायन्ते तेऽमुष्मिन्नादित्ये सृप्ताः
These sun's rays... they spread out from the sun and enter into these "nerves" and they spread out from these "nerves" and they gather into the sun.
This is an interesting idea of the connection of these channels of energy in the body to the channels of energy (i.e. rays) of the sun.
तद्यत्रैतत्सुप्तः समस्तः संप्रसन्नः स्वप्नं न विजानात्यासु तदा नाडीषु सृप्तो भवति तं न कश्चन पाप्मा स्पृशति तेजसा हि तदा संपन्नो भवति
When a person is asleep, self-contained, blissful, not dreaming, then he enters into these "nerves", he is not touched by sin, he is full of pure consciousness.
अथ यत्रैतदस्माच्छरीरादुत्क्रामत्यथैतैरेव रश्मिभिरूर्ध्वमाक्रमते...
Then when he leaves this body, he moves upwards via these rays.
तदेष श्लोकः -
शतं चैका च हृदयस्य नाड्यस्तासां मूर्धानमभिनिःसृतैका ।
तयोर्ध्वमायन्नमृतत्वमेति विष्वङ्ङन्या उत्क्रमणे भवन्ति ॥
In this regard here is a shloka:
Hundred and one are the heart's "nerves", of them one goes straight up to the head. By going out through that one, a person attains immortality. Exiting through others leads to different destinations.
So here we have the hints of the idea of the suṣumṇā nāḍī that's supposed to go straight to the crown of the head, where the sahasrāra is supposed to open up.
Another hint of this special point at the crown of the head is also seen in Aitareya Upanishad. After describing the allegorical creation of the human form, the Atman ponders about how he should enter this body (3.12):
स एतमेव सीमानं विदार्यैतया द्वारा प्रापद्यत सैषा विदृतिर्नाम द्वास्तदेतन्नान्दनम्
He split the crown of the head and entered through the opening. This opening is called vidṛti, and it is the place of ultimate bliss.
The Brhadaranyaka Upanishad (2.1.19) mentions the 72,000 nāḍīs that becomes the typical number in later texts.
अथ यदा सुषुप्तो भवति यदा न कस्यचन वेद हिता नाम नाड्यो द्वासप्ततिः सहस्राणि हृदयात्पुरीततमभिप्रतिष्ठन्ते ताभिः प्रत्यवसृप्य पुरीतति शेते
When a person is in deep sleep and is not aware of anything, he withdraws into these 72,000 "nerves" called hitā, which extend from the heart into the pericardium (or intestine), and he rests in the pericardium.
I recall reading or listening about this, but don't remember the source. In the Śrīvaiṣṇava tradition of Rāmānujācārya, the holy marks (ūrdhva tripuṇḍra, i.e. three vertical lines) applied on the forehead and other parts of the body are supposed to symbolize the three main nāḍīs - the middle red line the suṣumṇā, and the white lines on either side the iḍā and piṅgalā.