I’m not sure about the Democratic Party in the U.S. specifically, but if you want to say something like, can I have these broadly left wing commitments, or what would generally be considered left wing commitments (like supporting social safety net, mutual aid, poverty relief, worker solidarity, anti corporatism, anti racism/sexism/bigotry, LGBT rights, anti war/militarism, pro choice, etc.) and at the same time support a freed market and individual rights, including private property rights, then the answer is a resounding YES.
There is a rich tradition of individualists in the English, French, and American liberal tradition who have historically been supporters of economic freedom and at the same time supporters of left wing causes. Some have even embraced the term “socialist” in opposition to its association with totalitarian governments. Ayn Rand herself, while having a more right wing strain mostly in the later part of her life, has a distinct left wing legacy in her thought as well. Murray Rothbard and Ludwig von Mises also have left wing strains in their thought, Rothbard allied with the New Left in the mid 60’s against the Buckyleite right wing (Rand also broke the the same generation of neoconservatives), and Mises work on liberalism bears a book of the same name.
These writers embraced social cooperation and tolerance, upheld the rights of marginalized groups, fought vigorously against chattel slavery in the 19th century (Tucker, Spooner, Garrison), supported the burgeoning feminist movement and women’s rights (Patterson, Wilder, and Rand), and upheld a market economy against government planning and corporate capitalist management.
A lot of Rand’s ideas can appeal to someone would is broadly left wing, in this kind of way, that sort of believes that there is all kinds of injustice and massive power centers, corporate and governmental, that are unaccountable and pushing people around, and has a general pro reason and anti authoritarian bent. But there’s another kind of leftist, who is more establishment in attitude, the kind that would look up to, say Hillary Clinton or Elizabeth Warren, and has more of an attitude of, there’s all these things we want to change about the world, we have a blueprint of how we want things to be, and we’re going to use the government to forcibly and violently impose our blueprint on the world. This kind of statist liberal, no I don’t think can be compatible with a Randian or libertarian point of view.
For more info, there is a Wikipedia article on left libertarianism that you might want to check out. You might be interested in the book “Markets not Capitalism” edited by Gary Chartier and Charles Johnson, it’s online for free in .pdf form, or the website Alliance of the Libertarian Left, http://praxeology.net/all-left.htm .