Although the title of Deputy Leader of the Conservative Party has been applied to politicians in the past, the party currently has no deputy leadership position, and this is confirmed by the party's constitution, which contains no such provision, and this Guardian article from 2015, which notes the difference between the approaches of the Labour Party and the Conservatives on this issue. The Institute for Government, however, while noting that "The Conservative Party does not have a deputy leadership position", also states that "The party could appoint a temporary leader" during the leadership selection period.
If the leader of the party is unable to perform his duties, the duty of electing a new leader rests with the 1922 committee of backbench MPs, currently chaired by Sir Graham Brady MP. As I noted in my answer to this very similar question, there is nothing to stop the committee offering MPs a range of candidates, initiating a leadership contest, however it is also within their power to only nominate one candidate who would then be summarily confirmed as the leader of the party.
In conclusion then, as this has never happened before in modern times, and there is no specific provision for the scenario of the party leader's death in the party's constitution, it is impossible to be certain, however it seems to me there are four possible scenarios, which I will present in order of which I find to be most likely.
There is no acting leader position announced, and the 1922 committee speedily announces the new leader.
Dominic Raab MP, the PM's designated survivor, or another high-ranking cabinet member is declared as acting leader pending the formal selection of a new leader.
The Chairman of the 1922 committee is declared acting leader until the conclusion of the leadership contest/selection.
The Chairman of the party is announced as acting leader - unlikely, firstly because this is mostly an administrative role, and secondly because there are currently two co-chairs. If this were to be the scenario, the acting leader would probably be Amanda Milling MP, as Ben Elliot is not an MP. The party's constitution states that the leader of the party should be drawn from the party's MPs.
While it is not related to the specific question, given the current circumstances I feel that it is important to note that the positions of Prime Minister and leader of the largest party are not intrinsically linked, and just because a temporary leader might be chosen, this certainly does not mean that that temporary leader would necessarily become the PM.