Yes, a libertarian can vote Republican in elections if they choose to. A person may also simultaneously be a libertarian and a republican without contradiction.
Republicanism, in its true form, is a system of government where the power rests with the people, not a monarchy or oligarchy. In a true republic, citizens either represent themselves or have representatives. The United States is a Democratic Republic.
Libertarians who also identify as republicans typically believe in a minimalist system of government known as a Libertarian Republic, which has a stronger emphasis on individual sovereignty than a traditional republic.
Libertarianism is not synonymous with the Libertarian Party. However, many libertarians choose to register with the LP rather than the two major American political parties: The Republican Party or Democratic Party.
The Libertarian Party endorses a political system of government much like a Libertarian Republic.
Libertarianism is a philosophy that holds individual liberty as a core value. While the Republican Party does not share the exact same beliefs as libertarianism, they agree on many issues. A libertarian is free to vote for any candidate they wish.
Libertarians who register as republicans often do so to vote in primary elections, citing their need to cast ballots for candidates and specific issues that are important to them, such as gun liberty.
Libertarians like Ron Paul and Rand Paul represent the Republican Party. Their decision to run as republicans instead of with the Libertarian Party was likely a result of their desire to enter politics and be acknowledged by Congress, as the Libertarian Party does not share the same clout as the Republican Party. Still, they are libertarians.
Ultimately, republicans and libertarians, in the modern view, are not the same (Libertarians call for greater freedom and personal liberty than the Republican Party), but conservative libertarians tend to vote for republicans if there is no libertarian representation or if they feel a specific candidate has a better chance of making a political difference.
Conversely, a left-leaning libertarian may, for the same reasons, vote for democrats if they feel strongly about issues that the Democratic Party supports, although this is not as common.
For more information, see our Ultimate Guide to Libertarianism