Maternal immunity without prenatal vaccination?
tl;dr

If a woman was vaccinated when she was a child, will she be able to provide passive immunity to her children?

It depends on the vaccine. Generally, if the mother still has high levels of protective IgG  during the third trimester (especially if it's type 1 or 4), those antibodies will cross the placenta and provide passive immunity to the newborn. In whooping cough, though, good levels of newborn IgG require vaccination in the third trimester.
Maternal immunizations
Immunizations are given to adult women prior to conception, while pregnant, and after delivery to protect the mother from infection, protect the fetus from infection, protect the fetus from congenital anomalies associated with maternal infection, and to protect the newborn from infection through passive immunity and through decreased exposure to infected contacts. The exact recommendations differ for different vaccines and different levels of maternal risk. From the same recommendation, you can see that some vaccines are recommended only if maternal antibody titers are negative and some are recommended regardless of maternal antibody titers or prior immunization.
Passive immunity of the newborn
IgG undergoes active transport from the maternal circulation to the fetal circulation across the placenta.

The reference linked above is an excellent review of transplacental antibody transfer and the science that underpins maternal immunization recommendations for passive immunization of the newborn (especially in the case of whooping cough). Key points here include:

Newborn IgG levels correlate with maternal IgG levels (i.e., the active transport mechanism isn't saturated in most cases). This suggests that low IgG for a particular antigen in the mother will result in low IgG for that same antigen in the newborn.

Antibody transport begins as early as week 14, but both the total transport and the rate of transport increases over the length of the pregnancy. This suggests an hypothetical optimum time for an intervention that would temporarily boost the level of any antibody.


Pertussis example
Whooping Cough, or pertussis, the illness caused by Bordetella pertussis, an encapsulated bacteria, is quite dangerous in the newborn. A few years ago recommendations for vaccination against pertussis were updated to immunization in the third trimester, regardless of the mothers prior immunization status. This recommendation is based, in part, on the understanding of transplacental transfer of IgG described in the article I linked in the earlier section, and in part on data on the association between timing of maternal immunization and fetal, all in the context of increased pertussis case rates.
Shortly after these recommendations were updated, additional, more thorough data confirmed this recommendation, though it suggested a slightly narrower window for the ideal time of Tdap administration (from 27-36 weeks to 27-30 weeks).
An aside re: the relationship between passive immunity of the newborn and childhood immunizations
Passive immunity from antibodies transferred to the fetus during pregnancy declines pretty sharply after birth. By 8 months the maternal IgG contribution is effectively gone. This is passive immunity because only the products of B-cells (the antibodies) are transferred. Immunization of the child, on the other hand, provides active immunity, because they develop the capacity to produce their own antibodies. I'd refer you to the allergy and immunology chapter of Bernstein and Shelov's Pediatrics for Medical Students, a surprisingly good general reference.
How should we interpret the linked article in the OP
This article is a comment on a very interesting study in baboons. It doesn't present any new data, but does comment on epidemiological data from a natural Swedish cohort between the discontinuation of whole-cellular pertussis vaccine 1979 and introduction of acellular pertussis in 1996. They reference this data through a misinterpretation of yet another study that analyzed infection rates between 1986 and 2007. The misinterpretation is as follows:

The (1996) drop was pronounced in infants too young to be vaccinated (Fig. 1A), itself a clear indication of herd immunity.

This is incorrect. That figure represents the decrease in infection rates for infants under 1 year of age at the point when vaccine was re-introduced. This included infants who were vaccinated (at 3 and 5 months).
The data before the re-introduction of vaccine, e.g., in 1995, though, does provide a good natural experiment for the epidemiology of pertussis in a non-immunized and immunization remote population. It had been 16 years since the use of whole-cell pertussis and you can clearly see the waning immunity in the population incidence as a whole. If you are really interested in digging into these questions, this cohort is a good place to start. Sweden is very good at collecting important data, so it may be out there.