How many disciples were there of Yamunacharya sage and what are their names?
Yamunacharya had many disciples.
SrIvaishNava guruparamparai lists the shishyas of Yamunacharya.
Sishyas: periya nambi, periya thirumalai nambi, thirukkOshtiyur nambi, thirumAlaiAndAn, dheivavAriAndAn, vAnamAmalaiAndAn, IsvarAndAn, jeeyarAndAn ALavandhArAzhwAn, thirumOgUrappan, thirumOgur ninRar, dhEvaperumAL, mAranEri nambi, thirukkachi nambi, thiruvaranga perumAL arayar (sishya of maNakkAl nambi and son of ALavandhAr), thirukkurugUr dhAsar, vakulAbharaNa sOmayAjiyAr, ammangi, ALkoNdi, gOvindha dhAsar (who was born in vadamathurai), nAthamuni dhAsar (rAja purOhithar), thiruvarangathamman (rAja mahishi).
Out of these, there were five prime disciples. Yamunacharya recognised Ramanucharya's devotion towards Vishnu. He instructed his prime disciples to teach various aspects of Sri Vaishnava tradition to Ramanujacharya.
Mahapurna (Periya Nambi) - Periya Nambi brought Ramanujacharya to Yamunacharya. He performed samashrayanam to Ramanujacharya.
Srishaila Purna (Periya thirumalai nambi) - Periya Tirumalai Nambi is maternal uncle of Ramanujacharya. He gave the name "Ilayazhwar" to Ramanujacharya. He taught meaning of Ramayana which is regarded as Sharanagathi grantham in Sri Vaishnava tradition. Ramanujacharya developed interest in Venkateshwara due to influence of Tirumalai Nambi who also lived in Tirumala. We can find a temple dedicated to Shri Shaila Purna in Tirumala and Govindaraja swamy temple in Tirupati.
Goshtipurna (thirukkOshtiyur nambi) : Goshtipurna was given the responsibility of teaching the meaning of Tirumantram and Charama Shloka to Ramanujacharya by Yamunacharya. It was Goshti Purna who gave the name "Emberumanar" to Ramanujacharya.
Maaladhaara or Gyaana Purna (thirumAlai ANdAn) : thirumAlai ANdAn is given the responsibility of teaching the meanings of thiruvAimozhi.
thiruvarangapperumAL arayar - He is son of Alavandar i.e., Yamunacharya. He was an expert of music, dance and drama . He performed Arayar seva in front of Ranganatha in Sri Rangam. He was given the responsibility of teaching arulicheyal and Charamopayam (i.e., acharya nishtai the ultimate means). Ramanujacharya performed service for 6 months before asking him to teach anything.
Following is an image of Sri Vaishnava Guruparampara.