Vukobrat began playing at the local regional levels with various teams such as Dača Dimitrijević, and CSK Čortanovci.[2][3] In 2002, he joined the professional ranks to play in the Serbian League Vojvodina for a season with FK Inđija.[2] In the next season, he transitioned into a higher tier in the First League of Serbia and Montenegro with FK Srem and had a loan spell with FK Polet Novi Karlovci.[2] He would ultimately sign a permanent deal with Big Bull Bačinci in 2009 in the third division.[4][2] The following season Bačinci was promoted to the second tier where Vukobrat appeared in 32 matches and scored 1 goal.[5]
Donji Srem
After a season with Bačinci, he returned to Donji Srem in 2011 and helped the club win the Serbian League Vojvodina.[6][7][8] In his second stint with Srem, he assisted the club in securing promotion to the Serbian SuperLiga.[2] Throughout the season he played in 28 matches and recorded 2 goals.[5] He would re-sign with Srem during their run in the country's top tier.[9][10][11] In his debut season in the top division, he played in 23 games and recorded 1 goal.[5]
In 2013, he played abroad in the Oman Professional League with Al-Nasr SC.[12][4][3][13] After a season in the Middle East, he returned to the Serbian premier league with former club Srem for the 2014-15 season.[14][15] In his return to the top division, he played in 14 matches.[5] Vukobrat would continue playing in Serbia in the lower leagues with Radnički Sremska Mitrovica throughout the 2015-16 season.[16][17]
Asia
In the summer of 2016, he had another stint abroad this time in the Southeast Asian Dhivehi Premier League with United Victory.[18] He played with United Victory the following season in the regional Malé League.
Canada
After several seasons in the Maldives, he played abroad again in the southern Ontario-based Canadian Soccer League with Hamilton City.[19] In his debut season with Hamilton, he helped the club secure a playoff berth by finishing sixth in the First Division. The club would FC Ukraine United in the opening round of the postseason where Hamilton was eliminated from the competition.[20] He re-signed with Hamilton for the 2020 season.[21]
After the merger between Hamilton and Brantford Galaxy, he played with BGH City FC for the 2021 season.[22] Vukobrat returned for his second season with the hybrid team in 2022.[23] Throughout the 2022 campaign, the club secured a playoff berth by finishing sixth in the standings.[24] The team was defeated in the quarterfinal round of the postseason by Scarborough SC.[25] In 2023, he returned to his former club Hamilton as it became a single entity once again.[26]