According to received wisdom, pterosaurs died out 66 million years ago. However, despite the fossil record from which this conclusion is drawn, there have been reports in modern times of creatures that sound strangely like pterosaurs.
Bernard Heuvelmans drew our attention to African creatures which showed such characteristics. In Zambia, natives spoke of the kongamato which had a wingspan of 4'-7'. Dr J.P.F. Smithers described birds with billfuls of teeth which looked prehistoric in 1956 near Lake Bangweulu. An expert said they must have been shoebills. Shoebills, of course, don't have teeth. Perhaps these ones had been fitted with dentures.
Up in Kenya there have been reports of a similar creature, locally referred to as a Trappe pterosaur. Specimens of this creature are said to be found in the vicinity of Mount Kenya.
However, pterosaur-like creatures have also been reported from the United States. In 1855, Major H. Bell described such a creature which dwelt in Lake Elizabeth (California). It had batlike wings and did not smell too enticingly. Two ranchers are supposed to have shot such a beast in Arizona, according to the Tombstone Epitaph of 1890. Although the paper says they killed the creature, this appears to be an exaggeration, as does the huge wingspan the report gave it.
It is from around Raymondsville and Brownsville (Texas) that we must look for a number of reports.
1976: A creature was spotted near Brownsville. Some days later
a similar creature was seen near San Antonio. In the same
year A. Guajatado of Brownsville saw a horrid creature
with batlike wings and a beak. A couple named Ford also
reported a pterosaur like animal from the same area.
1983: J. Thompson saw a creature with featherless wings and 5'-6'
wingspan between Raymondsville and Brownsville.
In 1969 two men were boating on Lake Dewy (Michigan) when a huge flying creature came between them and the sunlight. The volant creature had leathery wings. Its wingspan was estimated at 14'.
In 1983 a pterosaur was reported from Thermal (California).
In 1998 a flying creature the size of a pickup truck sporting leathery wings was seen from a hospital window in Brookfield (Wisconsin).
In 2004 a pterosaur-like creature colored red and of leathery appearance with a wingspan of at least 30' was reported from Henderson (Kentucky).
This does not exhaust relevant reports.
The fossil record tells us these creatures do not still exist. The vast majority of creatures do not get fossilized, however, so I see no reason whole species could not go unrecorded.
For interested readers, there is a small volume entitled Live Pterosaurs in America by Jonathan Whitcomb (2009).