Yes. Because the permanents enter the battlefield at the same time, they are all there when the game checks for trigger conditions among them (two triggers for each land that entered due to two Tatyovas). This happens right away in the middle of the Vow's resolution, so quite a while before state based actions are checked. After thw Vow has finished resolving, state based actions send all but one of the Tatyovas to the graveyard (this is not destruction!), then all the waiting triggers go on the stack.
In this scenario, if one or more lands and a legendary permanent named Tatyova, Benthic Druid enter the battlefield under your control, the new Tatyova's and the old Tatyova's ability will each trigger once for each land that enters this way (C.R. 603.6a, 603.10). For example, if two lands enter this way, each Tatyova's ability will trigger twice, for a total of four times. After Kamahl's Druidic Vow resolves, all legendary permanents you control named Tatyova, Benthic Druid, except one of your choice, will go to the graveyard (C.R. 116.3b, 116.5, 704.5j).
The legend rule, like other state-based actions, happens only at discrete moments of the game (C.R. 704.5j, 704.3, 704.4; compare with C.R. 603.2), such as after a spell or ability on the stack (such as Kamahl's Druidic Vow) resolves (C.R. 116.3b, 116.5), so that, for example, an enters-the-battlefield ability of a legendary permanent (such as Tatyova) can trigger even if its controller controls another with the same name (C.R. 603.6a, 603.10). (Note that the legend rule doesn't "destroy" any permanents [C.R. 704.5j].) However, the legend rule and other state-based actions will be checked before any such enters-the-battlefield ability can go on the stack (C.R. 116.3b, 116.5), let alone get to resolve. See also this thread.
Situation, tatyova is on the field, cast vow, get lands and tatyova out. Do both tatyovas trigger before I destroy a copy?
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)
The legend rule, like other state-based actions, happens only at discrete moments of the game (C.R. 704.5j, 704.3, 704.4; compare with C.R. 603.2), such as after a spell or ability on the stack (such as Kamahl's Druidic Vow) resolves (C.R. 116.3b, 116.5), so that, for example, an enters-the-battlefield ability of a legendary permanent (such as Tatyova) can trigger even if its controller controls another with the same name (C.R. 603.6a, 603.10). (Note that the legend rule doesn't "destroy" any permanents [C.R. 704.5j].) However, the legend rule and other state-based actions will be checked before any such enters-the-battlefield ability can go on the stack (C.R. 116.3b, 116.5), let alone get to resolve. See also this thread.
EDIT (Apr. 22, 2019): Correctness edit.