Yes it is allowed for Hindus to worship Shaligrama.
Adishankaracharya referenced the Shaligrama being the abode of Vishnu twice. Once in in the Sixth Anuvaka of Taittirya Upanishad:
Now the śruti declares that the hridaya-ākāśa, the bright space in the
heart, is the proper place for the contemplation and immediate
perception of that Brahman whose limbs the other Gods are, just as the
sālagrāma stone is the proper place for the contemplation of Visnu.
Indeed, when contemplated there, that Brahman is immediately perceived
in all His attributes
And once in the Brahma Sutra Bhasyas (pg 178)
.'--Or else (taking brahmapura to mean gîvapura) we may understand the
passage to teach that Brahman is, in the city of the individual soul,
near (to the devout worshipper), just as Vishnu is near to us in the
The story of Vishnu becoming a Salagrama stone is narrated in Chapter 21 of the Prakriti Khanda of Brahma Vaivarta Purana where Vishnu becomes a stone due to the curse of Shankachuda's wife Tulasi. (The entire story is quite long so I'm linking only specific parts)
हे नाथ ते दया नास्ति पाषाणसदृशस्य च। छलेन धर्मभङ्गेन मम स्वामी त्वया
Tulasi said-O lord, you are heartless, having no compassion. You are
hard like a stone. You have defiled my chastity deceitfully and have
killed my husband.
पाषाणसदृशस्त्वं च दयाहीनो यत: प्रभो। तस्मात्पाषाणरूपस्त्वं भूवि देव
O lord, therefore you are stone-hearted and devoid of compassion. You
will therefore take the form of a stone on earth.
Lord Vishnu accepts this curse of Tulasi and assumes the shape of a mountain near the River Gandaki.
अहं च शैलरूपेण गण्डकीतीरसंनिधौ। अधिष्ठानं करिष्यामि भारते तव शापत:॥
Because of the curse pronounced by you I shall appear in the form of a
mountain near the river Gandaki.
वज्रकीटाश्च कृमयो वज्रदंष्टश्च तत्र वै। तच्छिलाकुहरे चक्र करिष्यन्ति
The insects will carve or cut the stones with their teeth besides my
cakra on the stones.
सद्यस्तद्देहजाता च बभूव गण्डकी नदी। हरेरंशेन शैलश्च तत्तीरे पुण्यदो
From the body of Tulasi at once appeared the river Gandaki. At the
bank of the said river a huge mountain appeared from the amsa of lord
Visnu, which was quite sacred for the humans.
कुर्वन्ति तत्र कीटाश्च शिलां बहूविधा मुने। जले पतन्ति या याश्व जलदाभाव
O Sage, the insects make many types of stones there. The stones of
dark complexion fall into the river daily.
As for the significance of the Shaligrama, its worship provides great merit and destroys sins enabling the devotee to ascend to Vaikuntha. The Shaligrama is also a residence of Vishnu and Lakshmi which is why it is auspicious.
शालग्रामशिला यत्र तत्र संनिहितो हरिः। तत्रैव लक्ष्मीर्वसति
यानि कानि च पापानि ब्रह्महत्यादिकानि च। तानि सर्वाणि नश्यन्ति
Wherever the stone of Salagrama is kept, lord Visnu with Laksmi also
resides there. By adoring the stone of Salagrama one is relieved of
all the sins including that of Brahmahatya.
पाठे चतुर्णा वेदानां तपसां करणे सति। तत्पुण्यं लभते नूनं
One derives the merit of reciting all the Vedas by worshipping the
Salagrama stone undoubtedly.
शालग्रामशिलातोयं नित्यं भुङते च यो नरः। सुरेप्सितं प्रसादं च
तस्य स्पर्श घ वाञ्छन्ति तीर्थानि निखिलानि च। जीवन्मुक्तो
महापूतोऽप्यन्ते याति हरेः पदम्।।८७।।
The one who sips the water of the Salagrama stone and also consumes
the prasada of the god, he remains without old age and is freed from
birth and death. All the sacred places become desirous of touching
such a person. Therefore the one having been relieved of birth and
death gets purified and moves on to the abode of Visnu.
And as Swami Vishwananda pointed out in the comments, Shaligrama is also worshipped by the Shaktas and one example of that is the consecration of the Bhavani at Pratapgad by Shivaji Maharaj.
The Siva Chhatrapati Bakhar of Sabhasad (pg 27) summarizes it thus:
Then the Raje loaded a cart with money, sent it to the Gandaki,
brought a stone of that river, made an image of Sri Bhavani with it
and established the goddess at Pratapgad.