Can foreigners become Hindu- Sanatani?
Yes they can. From the Garuda Purana:
Bhakti or devotion is of eight kinds. If this bhakti is found in a mleccha, he should be considered as a devout and knowing Brahmin. Giving things to him and taking things from him as gifts is proper. He should be treated with the same reverence as I myself (Lord Vishnu).
From the Mahabharata:
Indra said, 'The mlecchas should serve their mothers and fathers, their preceptors and other seniors, and recluses living in the woods. The mlecchas should also serve their kings. The duties and rites inculcated in the Vedas should also be followed by them. They should perform sacrifices in honour of the Pitris, dig wells, (and dedicate them to universal service), give water to thirsty travellers, give away beds and make other seasonable presents unto Brahmanas. Abstention from injury, truth, suppression of wrath, supporting Brahmanas and kinsmen by giving them their dues, maintenance of wives and children, purity, peacefulness, making presents to Brahmanas at sacrifices of every kind, are duties that should be practised by every person of this class who desire his own prosperity. Such a person should also perform all kinds of Paka-yajnas with costly presents of food and wealth. These and similar duties, O sinless one, were laid down in olden days for persons of this class. All these acts which have been laid down for all others should also be done by mlecchas, O king.'