The Chechen Republic was proclaimed in October 1991 following the collapse of the USSR and a referendum in the region. A constitution was adopted in March 1992. The constitution was semi-presidential. It is available in English here.
Art 71 (1) stated: “The President of Chechen Republic is elected for 5 years by the citizens of Chechen Republic by general and direct election of balloting.”.
Art. 79 states:
“(1) The Cabinet of Ministers of the Republic bears the responsibility before the President of the Republic. The newly formed Cabinet of Ministers presents for consideration of the Parliament of the Republic the program of forthcoming activity for the term of authorities.
(3) The Parliament of the Republic can express mistrust to the Cabinet of Ministers that entails its resignation. The decree on this question is accepted by the majority of votes of not less than two-thirds of general number of the members of the Parliament of the Republic.”
It is unclear how long the constitution was even nominally operational. In April 1992 President Dudayev began to rule by decree and in June 1993 parliament was dissolved. For its part, worldstatesmen.org states that the position of PM was abolished from April 1993 to June 1996. So, we can assume that there was some legal basis for semi-presidentialism for a short period in 1992-93 and then again from 1996. In January 1997 there were presidential and parliamentary elections and Aslan Aliyevich Maskhadov was elected as president.
In 2003, following the Russian military campaign, a new constitution was adopted that formally incorporated Chechnya within the Russian federation. The text is available here.
The wording of the translation of the 2003 constitution is a little confusing. The president is directly elected (Art. 65). There is a government and there is a position of chairman of the government (or prime minister). It appears as if the president’s nominee for prime minister has to be approved by parliament (Art. 83 (3) (a)). In addition, Art. 95 seems to suggest that the parliament can dismiss the government, but the wording is very unclear.
My understanding is that the Chechen government in exile, i.e., the one that refuses to acknowledge the incorporation of Chechnya as part of Russia, continued to follow the 1992 constitution even after 2003. However, in November 2007 Doku Umarov proclaimed himself Emir. So, the 1992 constitution is no longer semi-presidential.
As you can see, the situation is complicated. So, any clarification would be welcomed.