How does a non-Sri Vaishnava get initiated into Sri Vaishnavism? What Sri Vaishnava Acharyas are there in the world today who initiate people?
To be initiated into Sri Vaishnavism you need to do Samashrayanam, AKA Panchasamskaram, a ritual which among other things involves the branding of the shoulders with symbols of Vishnu’s Shankha and Chakra. Here is a video of the ritual. This ritual can be performed with any Sri Vaishnava Acharya. Here is a list of Sri Vaishnava Acharyas along with their websites. Note that some of them are affiliated with Mathams, large organizations, whereas others are Swayamacharya people who only have a small Thirumaligai or sacred house of residence.
- Vanamamalai Matham Jeeyar, affiliated with Vanamamalai Matham
- Chinna Jeeyar Swami, affiliated with Ubhaya Vedantacharya Peetham
- Sri Ranga Narayana Jeeyar, affiliated with Sri Ranga Narayana Jeeyar Matham in Sri Rangam, which was founded by Kuranarayana Jeeyar, son of Ramanujacharya’s cousin Siriya Govinda Perumal
- Thirukurungudi Jeeyar, affiliated with Thiru Jeeyar Mutt in Thirukurungudi
- Tirupati Periya Jeeyar, affiliated with Tirupati Periya Jeeyar Mutt
- Mudaliandan Swami, who is affiliated with Mudhaliandan Thirumaligai
- Doddacharya Swami, who is affiliated with Periyappangar Thirumaligai
- Koil Kandhadai Annan Swami, who is affiliated with Koil Kandhadai Annan Thirumaligai
- Dasarathi Swami, affiliated with Thirumani Thirumaligai
- Appan Swami, who is affiliated with Koil Kandhadai Appan Thirumaligai
- Bhavanachariar Swami, who is affiliated with Koil Kandhadai Bhavanachariar Thirumaligai
- Shuddhasattvam Annan Swami, who is affiliated with Koil Kandhadai Shuddhasattvam Sirupuliyur Annan Thirumaligai
- Elayavilli Achan Swami, who is affiliated with Koil Kandhadai Elayavilli Varadhachariar Thirumaligai
- Ahobilam Matham Jeeyar, affiliated with Ahobilam Matham
- Parakala Matham Jeeyar, affiliated with Parakala Matham
- Sri Rangam Andavan, affiliated with a Sri Rangam Andavan Ashramam
- Poundarikapuram Andavan, affiliated with Poundarikapuram Andavan Ashramam
- Sri Krishna Desika Jeeyar Swami, affiliated with Sri Krishna Ashram
- Karunakarachariar Swami, who is affiliated with Nadadur Ammal Thirumaligai
- Navalpakkam Swami, who is affiliated with Navalpakkam Thathachariar Thirumaligai
EDIT: Regarding Thenkalais and Vadakalais, every Jeeyar is either a Thenkalai or a Vadakalai. So by choosing to receive Samashrayanam from a particular Jeeyar, a person is joining the sub-sect of Sri Vaishnavism that that Jeeyar belongs to.