I don't pretend that the examples that follow are exhaustive, and I'll gladly complement this answer with input from other answers or comments.
Mohamed Morsi won what is considered the first democratic presidential election of the country in 2012. He is a leader of the Muslim Brothers (and of its political branch, the Freedom and Justice Party) and often refers to sharia. His first major move as President was:
Morsi sought to influence the drafting of a new constitution of Egypt, favoring a constitution that protects civil rights and enshrines Islamic law.
In November 2012 this project of a new constitution was revealed and heavily criticized, but more so because it concentrated powers in the hands of the President than because of its references to sharia.
The Constitution was adopted by referendum on 15 and 22 december 2012. It gained 64% of the votes, but with a poor turnout of 33%.
Article 2, makes "the principles of Islamic law the main source of legislation," a statement defining the relationship between Islam and Egyptian law, essentially unchanged from Egypt's old constitution. At the urging of Islamists, another article was added to the constitution strengthening the relationship, defining the "principles of Shariah" in the terms of Muslim Sunni jurisprudence i.e. "evidence, rules, jurisprudence and sources" accepted by Sunni Islam. Liberals fear "Islamic punishments for things like theft, adultery, and blasphemy are not far behind".
In July 2013, a military coup ended Morsi's Presidency. A new Constitution was adopted in January 2014, also by referendum. It doesn't seem to enforce sharia:
Under the [new] constitution, there is a guarantee of equality between the sexes and an absolute freedom of belief, but Islam is the state religion
Algeria held multipartite legislative elections in 1991.
The country was no paragon of democracy at the time, but the elections themselves were a democratic process.
Islamic Salvation Front got a huge advance after the first round, but a military coup stopped the election process at that point (leading to 10 years of civil war). Islamic Salvation Front's program included the establishment of an Islamic khalifate, so I assume that, once elected, they would have introduced sharia.
We can debate how effective a democracy Nigeria is at the federal level, and at the state level it is probably not a great one, especially in the North of the country. The institution of sharia in 9+3 states should probably not count as "introduced by democratic means", yet it happened in a merely democratic country.
Sharia in the world
This map documents countries in which sharia plays a role as per today. A good base for further research: has its introduction come from a democratic process in any of them ?