Nearly alone among South American nations, Venezuelan law leaves designation of a vice president until after the new leader is sworn in. It did not even have a vice president until 1999, when the office was created in a new constitution promoted by Chavez.
Campaigning against youthful challenger Henrique Capriles, Chavez seldom mentions his health problems, which over the past 15 months have required frequent trips to Cuba for three surgeries, chemotherapy and radiation treatments. When asked about his health last week, Chavez said his last checkup in June showed he was cancer-free and "everything will be fine."
Still, even some loyal Chavez supporters who plan to vote for him on Oct. 7 acknowledge feeling uneasy about the uncertainty.
"Who would take his place? Nobody knows, and that's makes me worry," said Maria Lovera, a street vendor selling household cleaning supplies on a street corner.
"I love Chavez and I want him to remain in power for many more years, but I must confess that some people like me have suspicions that he hasn't told us the whole truth about his cancer," Lovera said. "The possibility of him getting sick again and having to step down exists."
Some say identifying a vice president now could cause divisions within Chavez's movement by favoring one faction over another.
And because none of Chavez's possible vice presidential candidates have anywhere near his popularity, choosing one might alienate some sectors within his movement or prove less appealing to voters still on the fence.
"He is aware that if he appoints the VP (ahead of the vote), that would trigger divisions and increase uncertainties," said Diego Moya-Ocampos, an analyst with the consulting firm IHS Global Insight in London.
Capriles has not named a likely vice president either, though the issue is less pressing because there are no questions about the 40-year-old's health.
"Capriles, like Chavez wants to make this a contest between the two figures," said Miguel Tinker Salas, a Latin American studies professor at Pomona College in Claremont, California. For both candidates, he said, to name a vice president now "would detract from this strategy."
Under Chavez, the vice presidency has been an office of limited authority. In fact, the country's constitution doesn't even specify how soon a president must name a vice president after winning office.
Chavez has delegated little to his vice presidents over the years, instead using his deputy as one in a team of aides who largely carry out his instructions. Even when he was forced to scale back his work due to taxing cancer treatments in Cuba, Chavez delegated only minimal administrative duties to current Vice President Elias Jaua, whom he appointed in 2010, the seventh man to hold that post during the socialist leader's 13 years in office.
Venezuela's Constitution says that if a president-elect dies before taking office, new elections would be called. If a president dies during the first four years of the term, the vice president would temporarily take over, but a new election would be held. If a president dies during the last two years of the six-year term, the vice president would finish the term and the next election would be held as planned.
In most South American countries vice presidents are elected at the same time as the top of the ticket. In Chile, however, presidential candidates run for office alone, but once in office name an interior minister who serves as vice president when necessary.
As for which ally Chavez might pick as his deputy if re-elected, speculation in recent months has ranged widely across figures such as Foreign Minister Nicolas Maduro and National Assembly President Diosdado Cabello.
Throughout his presidency, Chavez's closest confidants have been characterized by their lack of an independent following, a relative lack of charisma and military-like allegiance to their leader, a former army paratroop commander.
Moya-Ocampos said the vice-presidential mystery is by design. By saving any talk of his choice for later, he said, Chavez aims to send a simple message, "that there is no clear successor."