How many people living in the U.S. today can trace their lineage to living in the U.S. in the 1700s or earlier?

I don't know about the 1700's in particular, but there has been some research down on what proportion of Americans can trace their descent to people on the Mayflower. A New York Times article from 2001 says:

And, perhaps in defiance of the laws of mathematics, 25 percent of the American people today believe they are descended from the Pilgrims, according to a survey done last November for the General Society of Mayflower Descendants.

The General Society of Mayflower Descendants have also produced their own estimates. According to this Boston Globe article from 2017:

Today, there are an estimated 10 million living Americans and as many as 35 million people worldwide descended from that resilient little group, said Lea Filson, governor general of the Plymouth-based General Society of Mayflower Descendants. They include presidents, poets and celebrities.

This 10 million figures means that about 3% of Americans are Mayflower descendants.

Another way to look at this would be to look at what percentage of people claim English or African American descent (Congress outlawed the importation of slave after January 1st, 1808). Between 1980 and 2000, the Census asked about "ancestry" on their long form census. In 2000, you were allowed to specify multiple ancestries.

You have detailed write up from the census here. Here are the results of the groups whose largest waves of immigration were before 1800:

Group	Percent in 2000
African-American	8.8%
English	8.7%
"American"	7.8%
American Indian	2.8%
Scottish	1.7%
Dutch	1.6%
Scots-Irsh	1.4%
Total	32.8%

But this also shows the futility of this method of estimating. Most "Americans" are assumed to be English or Scots-Irish. Also, obviously, recent English, Dutch, African, etc. immigration is misclassified with this method. I have cousins who are of English descent, but they’re first generation immigration. Likewise, older immigration, particularly Irish and perhaps German, from groups whose peak of immigration occurred after 1800 are omitted entirely. Some percentage of Mexican non-immigration is also missed, as in some places the U.S. border came to Mexicans rather than Mexicans coming to the U.S. I once asked a Chicana friend when her ancestors came to American and she said, "Well, on my mom's side, her parents immigrated, but my dad's side is from East Texas. The U.S. came to them, you know, they didn't come to the U.S. They'd been on the same land forever before my dad came to L.A." This was only among the top-15 largest groups in America, some other groups (i.e. Cajuns, Welsh, even Sephardic Jews like poet Emma Lazarus and Supreme Court Justice Benjamin Cardozo) also heavily immigrated to the future United States sometime before 1800.

Further, in answering all questions like this, people almost always choose to emphasize one side of their family line over all the others. I would probably answer this question that my family was "Jewish" or "German" or "German-Jewish", which is true, but incomplete. That's only one side of my family line. On the other side, I am actually all WASP, a combination of recent (20th century) English migration and much older (likely to the 1700's) Welsh, Scots-Irish, and English ancestry, as well as vague rumors of Native American descent (Cherokee) that don't actually show up on the family trees we have going back to the 1700’s. This is common. My brother-in-law, for instance, would say he's "Italian", but on his mother's side he's actually French-Canadian and Belgian. The sociologist Mary Waters has written a fantastic book about this process called Ethnic Options.

Mary Waters's book is a combination of raw census data and detailed interviews. She notices a few trends: 1) people generally emphasize their fathers' sides more than their mothers, especially when their last name is ethnically identifiable, 2) if there's a "stigmatized identity" in there, that is more likely to be omitted (for example, Russian identity during the time of the Soviet Union), 3) "exotic" identities tend to be emphasized over "common" ones, such that "Irish", "Italian", and almost everything else gets emphasized over "English". With race, this is even stronger: many people who are "half-Asian" end up simply identifying as "Asian", even if some of their ancestors came over on the Mayflower, 4) if people practice a mostly ethnically defined religion, they will more likely think of themselves as that ethnicity. A high school friend of mine, for instance, was English-German (with a clearly German last name) on her father’s side, but went to a Greek Orthodox Church and thought of herself primarily as “Greek”.

All this makes me think that estimating based purely the census answers to ancestries doesn't get you a particularly meaningful number. A large proportion of Americans simply just don't know with much detail where all of their ancestors came from. I'd put some stock in that the Mayflower survey, where 25% of Americans think they are descended from people who were aboard the Mayflower, not necessarily meaning their forebears were on the literal Mayflower but that they have at least some long, English roots in the American territory. That's, of course, mostly White Americans, so we'd need to add at least African-Americans to that number, which would lead us to say that 1/3 is not horrible as a quick low estimate, which is also roughly the same as we got from the very, very rough Census-ancestry based estimate (which, again, assume that essentially everyone with English heritage has at least one ancestor here before 1800). Based on things like Mary Waters's work where she found "English" ancestries tend to be omitted when narrating one's own history (and therefore, over generation forgotten), I would tend to believe that the real proportion of people who are descended at least in part from people arrived in the country before 1800 is likely considerably higher than 1/3.