Do Secular Institutes take vows?
Some do! Yes, it is commonly perpetrated that secular institutes do not take vows, but in fact, there are some that do. While the foundation document for Secular Institutes, Provida Mater Ecclesia, did not necessitate public vows for Secular Institutes, neither did the revised Code of Canon Law prohibit the possibility of public vows. So some do, and some don't.
In norms common to ALL Institutes of Consecrated Life, the church teaches that "Life consecrated through the profession of the evangelical counsels is a stable form of living, in which the faithful follow Christ more closely under the action of the Holy Spirit, and are totally dedicated to God, who is supremely loved...The state of persons who profess the evangelical counsels in these institutes belongs to the life and holiness of the Church..." Canon Law, #573, #574 Key here, is that consecrated life is marked by "profession of the evangelical counsels". Each institute determines the manner in which this happens.
Then, with respect to Secular Institutes in particular, it teaches that "the constitutions are to establish the sacred bonds by which the evangelical counsels are undertaken in the institute." Canon Law, #712.
So what is a sacred bond? Most often, in the context of consecrated life, it is understood as an oath or a promise. However, marriage is constituted by a sacred bond, the nature of which is deemed a "vow". So a sacred bond does not exclude a vow.
But in secular institutes, aren't vows "private vows"?
Irrespective of the "nature" of the sacred bond - vows, promise, oath, ..., for an Institute of Consecrated Life, to be "official", such vows (or other form of sacred bonds) are officially "received" by a delegate of the Institute in the name of the Church and made according to the constitutions of the Institute. Such conditions constitute the nature of a public vow - see Canon 1192. However, members of Secular Institutes who live their consecration "in the world" usually do so with a degree of approved and appropriate "discretion". Such discretion is often misinterpreted as rendering their consecration of a private nature, when in fact it has public, i.e. "official", recognition within the church.
So what Secular Institutes take vows?
The Company of St Ursula, Secular Institute of St Angela Merici, gives its members the choice of vow or promise as their sacred bond for professing the evangelical counsels. Notre Dame de Vie is another secular Institute which professes the evangelical counsels through the sacred bond of a vow. There may well be others.