BORI critical edition of Mahabharata is available online on sanskritdocuments.org
This article on hindustantimes.com says that BORI has ancient manuscripts of Rigveda.
Is there a BORI critical edition of 'rigveda' which is available online?
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This is not a critical edition by BORI. They have manuscripts with Sayana's commentary which were referred to by German scholar Max Müller to write a translation on the Rigveda. They have those manuscripts. The same news paper article link you have added in the question contains the following:
BORI’s M G Dadhphale said: “These were the very manuscripts referred to by German scholar Max Mueller, who first translated the Vedas into English. These manuscripts link our oldest text with the modern world.”
BORI official website contains the same. From their Manuscript department:
Accession No. : 5/1875-76
Material : Birch Bark
Uniqueness / Pecularity : The manuscript was used by Prof. F. Max Müller for the edition of the Rgveda with Sayana's commentary. The manuscript is a beautiful specimen of the old style Manuscript of Kashmir.
Translations of Rigveda Samhita by Max Müller are available on the Wayback machine in different volumes. So, we can search for them on that site.
According to this UNESCO Memory of the World Register nomination, there's a critical edition of the Ṛgveda (Saṃhita) published by Vaidika Samshodhana Mandala (Vedic Research Institute), Pune, India.
Out of the total number of 28,000 manuscripts housed at the Bhandarkar Oriental Research Institute, Pune, the 30 manuscripts of the Rigveda form a valuable part of the collection. These manuscripts evince several unique features in terms of scripts, accentuation marks and support material used, among others. Even the pioneering Indologist, Prof. F. Max Müller, has referred to one of these Rigveda manuscripts currently at the Institute. The material in this collection of Rigveda manuscripts was also used to prepare the well known Critical Edition of the Rigveda by the Vaidika Samshodhana Mandala, a premier institute in Pune for Vedic Studies. These manuscripts are of a high value as unique examples of the intellectual and cultural heritage not only of India, but of the world.
There are 30 manuscripts of Rigveda at the Institute [BORI], collected from different parts of India like Kashmir, Gujarat, the then Rajaputana, Central Provinces etc. They are written in Sharada, Devanagari and Devanagari with Prishthamatra and the material used for writing is birch bark as well as paper. The oldest of these manuscripts is dated 1464 A. D. It may be pointed out that manuscript no. 5/1875-76 was used by Prof. Max Müller for his edition of the Rigveda with Sayana’s commentary.
Out of these 30 manuscripts, 9 contain only the basic text. There are 5 manuscripts which contain the text along with Padapathas in which each meaningful unit of the words is separated from euphonic combination. The text of the Padapatha helps us in identifying every word separately and this ultimately helps us understand the meaning of the words to some extent.
As it says above, the Vedic Research Institute used the manuscripts available at BORI to create their critical edition.
I'm guessing the book Rgveda-Samhita with the Commentary of Sayana (In Five Volumes): Sanskrit Only available at Exotic India is the same critical edition. One of the volumes is available at archive.org.